Algebraic Topology on Polyhedrafrom Linear Finite Elements

Max Wardetzky

(with K Hildebrandt and K Polthier)

Free University Berlin amp

DFG Research Center ldquoMatheonrdquo

Oberwolfach March 6 2006

Letrsquos start with a concrete problem

Problem Compute space of harmonic vector fieldsfor meshes of genus gt 0

bull no vorticesbull no sinkssources

No maybeNo

Letrsquos start with a concrete problem

Problem Compute space of harmonic vector fieldsfor meshes of genus gt 0

Letrsquos start with a concrete problem

Problem Compute space of harmonic vector fieldsfor meshes of genus gt 0

locally integrable byharmonic fcts

Applications Mesh parameterization (cf GuYau) texture mapping shape matching shape morphing hellip

Harmonic vector fields

Strategybull discretize deRham complexbull get model for cohomology Hsup1bull discretize Hodge decompositionbull get harmonic vector fieldsbull compute basis of harmonic vector fields

Problem Compute space of harmonic vector fieldsfor meshes of genus gt 0

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Review smooth deRham complex

functions 1-forms 2-forms

(Mg) smooth surface Its deRham complex is

d is Cartan outer differential (no metric)

Review smooth deRham complex

functions 1-forms 2-forms

(Mg) smooth surface Its deRham complex is

This is a chain complex

d is Cartan outer differential (no metric)

Review smooth deRham complex

functions 1-forms 2-forms

vector fields

(Mg) smooth surface Make use of metric g

Review smooth deRham complex

functions 1-forms 2-forms

(Mg) smooth surface Make use of metric g

vector fields

sharp Hodge star

Review smooth deRham complex

vector fields

functions 1-forms 2-forms

(Mg) smooth surface Make use of metric g

Review deRham cohomology

First deRham comology of smooth surface (Mg)from grad-curl chain complex

Review deRham cohomology

encodes global topology

First deRham comology of smooth surface (Mg)from grad-curl chain complex

Local integrability iff locally

Review deRham cohomology

First deRham comology of smooth surface (Mg)from grad-curl chain complex

encodes global topology

Review deRham cohomology

First deRham comology of smooth surface (Mg)from grad-curl chain complex

Strategy for discretization

Local integrability iff locally

encodes global topology

1st step2nd step

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Discrete deRham

space of continuous PL functions on polyhedron

Discrete gradient

Discrete deRham

space of continuous PL functions on polyhedron

Discrete gradient

bull Gradients are piecewise constant

Discrete deRham

space of continuous PL functions on polyhedron

Discrete gradient

bull Gradients are piecewise constantbull tangential components are equal

along any edge

Discrete deRham

space of continuous PL functions on polyhedron

bull Gradients are piecewise constantbull tangential components are equal

along any edge

Local integrability iff locally

Discrete gradient

Discrete deRham

Discrete curl

(tangential jump)

ifftangential components are equalalong any edge

Discrete deRham

(tangential jump)

PL functionson vertices

PL functionson edge midpoints

PC vectorfields

Discrete deRham complex

ifftangential components are equalalong any edge

Discrete deRham

Lagrangeelement

Crouzeix-Raviartrsquo73

Discrete deRham

Lagrangeelement

Crouzeix-Raviartrsquo73

If then complex is exact in middle comp

bull ArnoldFalkrsquo89 ldquoA uniformly accurate FEM for Reissner-Mindlin Platerdquobull PolthierPreussrsquo03 ldquoIdentifying Vector Field Singularitieshelliprdquo

Discrete deRham

In fact complex yields correct cohomology

(dim = 2genus)

(not surprising - think simplicial = closed simplicial 1-forms)

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Discrete Hodge for closed meshes

(adjoint complex with respect to Lsup2 inner products)

Discrete Hodge for closed meshes

Hodge decomposition is simple linear algebra (since dim is finite)(B Eckmannrsquo44 ldquoHarmonische Funktionenhelliprdquo)

Discrete Hodge for closed meshes

harmonic vector fieldsDiscrete Hodge decomposition

Discrete Hodge for closed meshes

harmonic vector fieldsDiscrete Hodge decomposition

Lemma Decomposition only depends on choice of spaces and inner product on It does neither

depend on inner products on and nor on choice of operators

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Discrete Harmonic vector fields

dim = 2genus

Discrete Harmonic vector fields

Solve sparse systemfor u and v

dim = 2genus

Get method for computing

Computing harmonic fields

dim = 2genus

1 Init H = 0 2 Pick random PC vector field X3 Compute potential and co-potential of X4 Compute harmonic part h of X5 h in H IF TRUE goto 2 ELSE H = span(Hh)6 dim(H) = 2genus IF TRUE exit ELSE goto 2

Complexity (4genus) sparse systems (because odds that step 5 returns TRUE are zero)

Las Vegas algorithm for computing harmonic fields

Computing harmonic fields

Harmonic fields cont

Harmonic fields cont

no tangential jump

X is closed

Harmonic fields cont

no tangential jump

X is co-closed

X is closed

Harmonic fields amp harmonic1-forms

(no tangential jump)

X is closed ie X corresponds tosimplicial 1-form

X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

this 1-form is co-closed(condition at vertices)X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

this 1-form is closed(condition at faces)X is pcw cnst

this 1-form is co-closed(condition at vertices)X is co-closed

this 1-form is co-closed(condition at vertices)

this 1-form is closed(condition at faces)X is pcw cnst harmonic 1-froms on graphs

cf GortlerGotsmanThurston

X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

(watch out cotan edge weightscould be negative cf BobenkoSpringborn forgetting non-negative weights by intrinsic edge flips)

Discrete Hodge Star

Complex multiplication (rotation by )

Discrete Hodge Star

X has no normal jump across edges

Complex multiplication (rotation by )

hellipdoes not take to itself

Because

X has no tangential jump across edges

X would have to be globally continuous

Discrete Hodge Star

Complex multiplication (rotation by )

Hodge star

hellipin other words Hodge star of X is harmonic part of J(X)

hellipdoes not take to itself

Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

(image are piecewise linear forms)

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

Adjoint operator

yields Whitney-Hodge decomposition

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57) by construction

deRham map DodziukPatodirsquo76

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of area

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Convergence of Hodge decomposition

Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)

bull This remains valid for variable (and converging) metrics

bull FEM-Whitney relations

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Thank you

• Algebraic Topology on Polyhedrafrom Linear Finite Elements
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Harmonic vector fields
• Contents
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Contents
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Contents
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Contents
• Discrete Harmonic vector fields
• Discrete Harmonic vector fields
• Computing harmonic fields
• Computing harmonic fields
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Discrete Hodge Star
• Discrete Hodge Star
• Discrete Hodge Star
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• Contents
• Convergence What can go wrong
• Convergence What can go wrong
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition

Letrsquos start with a concrete problem

Problem Compute space of harmonic vector fieldsfor meshes of genus gt 0

bull no vorticesbull no sinkssources

No maybeNo

Letrsquos start with a concrete problem

Problem Compute space of harmonic vector fieldsfor meshes of genus gt 0

Letrsquos start with a concrete problem

Problem Compute space of harmonic vector fieldsfor meshes of genus gt 0

locally integrable byharmonic fcts

Applications Mesh parameterization (cf GuYau) texture mapping shape matching shape morphing hellip

Harmonic vector fields

Strategybull discretize deRham complexbull get model for cohomology Hsup1bull discretize Hodge decompositionbull get harmonic vector fieldsbull compute basis of harmonic vector fields

Problem Compute space of harmonic vector fieldsfor meshes of genus gt 0

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Review smooth deRham complex

functions 1-forms 2-forms

(Mg) smooth surface Its deRham complex is

d is Cartan outer differential (no metric)

Review smooth deRham complex

functions 1-forms 2-forms

(Mg) smooth surface Its deRham complex is

This is a chain complex

d is Cartan outer differential (no metric)

Review smooth deRham complex

functions 1-forms 2-forms

vector fields

(Mg) smooth surface Make use of metric g

Review smooth deRham complex

functions 1-forms 2-forms

(Mg) smooth surface Make use of metric g

vector fields

sharp Hodge star

Review smooth deRham complex

vector fields

functions 1-forms 2-forms

(Mg) smooth surface Make use of metric g

Review deRham cohomology

First deRham comology of smooth surface (Mg)from grad-curl chain complex

Review deRham cohomology

encodes global topology

First deRham comology of smooth surface (Mg)from grad-curl chain complex

Local integrability iff locally

Review deRham cohomology

First deRham comology of smooth surface (Mg)from grad-curl chain complex

encodes global topology

Review deRham cohomology

First deRham comology of smooth surface (Mg)from grad-curl chain complex

Strategy for discretization

Local integrability iff locally

encodes global topology

1st step2nd step

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Discrete deRham

space of continuous PL functions on polyhedron

Discrete gradient

Discrete deRham

space of continuous PL functions on polyhedron

Discrete gradient

bull Gradients are piecewise constant

Discrete deRham

space of continuous PL functions on polyhedron

Discrete gradient

bull Gradients are piecewise constantbull tangential components are equal

along any edge

Discrete deRham

space of continuous PL functions on polyhedron

bull Gradients are piecewise constantbull tangential components are equal

along any edge

Local integrability iff locally

Discrete gradient

Discrete deRham

Discrete curl

(tangential jump)

ifftangential components are equalalong any edge

Discrete deRham

(tangential jump)

PL functionson vertices

PL functionson edge midpoints

PC vectorfields

Discrete deRham complex

ifftangential components are equalalong any edge

Discrete deRham

Lagrangeelement

Crouzeix-Raviartrsquo73

Discrete deRham

Lagrangeelement

Crouzeix-Raviartrsquo73

If then complex is exact in middle comp

bull ArnoldFalkrsquo89 ldquoA uniformly accurate FEM for Reissner-Mindlin Platerdquobull PolthierPreussrsquo03 ldquoIdentifying Vector Field Singularitieshelliprdquo

Discrete deRham

In fact complex yields correct cohomology

(dim = 2genus)

(not surprising - think simplicial = closed simplicial 1-forms)

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Discrete Hodge for closed meshes

(adjoint complex with respect to Lsup2 inner products)

Discrete Hodge for closed meshes

Hodge decomposition is simple linear algebra (since dim is finite)(B Eckmannrsquo44 ldquoHarmonische Funktionenhelliprdquo)

Discrete Hodge for closed meshes

harmonic vector fieldsDiscrete Hodge decomposition

Discrete Hodge for closed meshes

harmonic vector fieldsDiscrete Hodge decomposition

Lemma Decomposition only depends on choice of spaces and inner product on It does neither

depend on inner products on and nor on choice of operators

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Discrete Harmonic vector fields

dim = 2genus

Discrete Harmonic vector fields

Solve sparse systemfor u and v

dim = 2genus

Get method for computing

Computing harmonic fields

dim = 2genus

1 Init H = 0 2 Pick random PC vector field X3 Compute potential and co-potential of X4 Compute harmonic part h of X5 h in H IF TRUE goto 2 ELSE H = span(Hh)6 dim(H) = 2genus IF TRUE exit ELSE goto 2

Complexity (4genus) sparse systems (because odds that step 5 returns TRUE are zero)

Las Vegas algorithm for computing harmonic fields

Computing harmonic fields

Harmonic fields cont

Harmonic fields cont

no tangential jump

X is closed

Harmonic fields cont

no tangential jump

X is co-closed

X is closed

Harmonic fields amp harmonic1-forms

(no tangential jump)

X is closed ie X corresponds tosimplicial 1-form

X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

this 1-form is co-closed(condition at vertices)X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

this 1-form is closed(condition at faces)X is pcw cnst

this 1-form is co-closed(condition at vertices)X is co-closed

this 1-form is co-closed(condition at vertices)

this 1-form is closed(condition at faces)X is pcw cnst harmonic 1-froms on graphs

cf GortlerGotsmanThurston

X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

(watch out cotan edge weightscould be negative cf BobenkoSpringborn forgetting non-negative weights by intrinsic edge flips)

Discrete Hodge Star

Complex multiplication (rotation by )

Discrete Hodge Star

X has no normal jump across edges

Complex multiplication (rotation by )

hellipdoes not take to itself

Because

X has no tangential jump across edges

X would have to be globally continuous

Discrete Hodge Star

Complex multiplication (rotation by )

Hodge star

hellipin other words Hodge star of X is harmonic part of J(X)

hellipdoes not take to itself

Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

(image are piecewise linear forms)

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

Adjoint operator

yields Whitney-Hodge decomposition

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57) by construction

deRham map DodziukPatodirsquo76

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of area

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Convergence of Hodge decomposition

Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)

bull This remains valid for variable (and converging) metrics

bull FEM-Whitney relations

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Thank you

• Algebraic Topology on Polyhedrafrom Linear Finite Elements
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Harmonic vector fields
• Contents
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Contents
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Contents
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Contents
• Discrete Harmonic vector fields
• Discrete Harmonic vector fields
• Computing harmonic fields
• Computing harmonic fields
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Discrete Hodge Star
• Discrete Hodge Star
• Discrete Hodge Star
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• Contents
• Convergence What can go wrong
• Convergence What can go wrong
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition

Letrsquos start with a concrete problem

Problem Compute space of harmonic vector fieldsfor meshes of genus gt 0

Letrsquos start with a concrete problem

Problem Compute space of harmonic vector fieldsfor meshes of genus gt 0

locally integrable byharmonic fcts

Applications Mesh parameterization (cf GuYau) texture mapping shape matching shape morphing hellip

Harmonic vector fields

Strategybull discretize deRham complexbull get model for cohomology Hsup1bull discretize Hodge decompositionbull get harmonic vector fieldsbull compute basis of harmonic vector fields

Problem Compute space of harmonic vector fieldsfor meshes of genus gt 0

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Review smooth deRham complex

functions 1-forms 2-forms

(Mg) smooth surface Its deRham complex is

d is Cartan outer differential (no metric)

Review smooth deRham complex

functions 1-forms 2-forms

(Mg) smooth surface Its deRham complex is

This is a chain complex

d is Cartan outer differential (no metric)

Review smooth deRham complex

functions 1-forms 2-forms

vector fields

(Mg) smooth surface Make use of metric g

Review smooth deRham complex

functions 1-forms 2-forms

(Mg) smooth surface Make use of metric g

vector fields

sharp Hodge star

Review smooth deRham complex

vector fields

functions 1-forms 2-forms

(Mg) smooth surface Make use of metric g

Review deRham cohomology

First deRham comology of smooth surface (Mg)from grad-curl chain complex

Review deRham cohomology

encodes global topology

First deRham comology of smooth surface (Mg)from grad-curl chain complex

Local integrability iff locally

Review deRham cohomology

First deRham comology of smooth surface (Mg)from grad-curl chain complex

encodes global topology

Review deRham cohomology

First deRham comology of smooth surface (Mg)from grad-curl chain complex

Strategy for discretization

Local integrability iff locally

encodes global topology

1st step2nd step

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Discrete deRham

space of continuous PL functions on polyhedron

Discrete gradient

Discrete deRham

space of continuous PL functions on polyhedron

Discrete gradient

bull Gradients are piecewise constant

Discrete deRham

space of continuous PL functions on polyhedron

Discrete gradient

bull Gradients are piecewise constantbull tangential components are equal

along any edge

Discrete deRham

space of continuous PL functions on polyhedron

bull Gradients are piecewise constantbull tangential components are equal

along any edge

Local integrability iff locally

Discrete gradient

Discrete deRham

Discrete curl

(tangential jump)

ifftangential components are equalalong any edge

Discrete deRham

(tangential jump)

PL functionson vertices

PL functionson edge midpoints

PC vectorfields

Discrete deRham complex

ifftangential components are equalalong any edge

Discrete deRham

Lagrangeelement

Crouzeix-Raviartrsquo73

Discrete deRham

Lagrangeelement

Crouzeix-Raviartrsquo73

If then complex is exact in middle comp

bull ArnoldFalkrsquo89 ldquoA uniformly accurate FEM for Reissner-Mindlin Platerdquobull PolthierPreussrsquo03 ldquoIdentifying Vector Field Singularitieshelliprdquo

Discrete deRham

In fact complex yields correct cohomology

(dim = 2genus)

(not surprising - think simplicial = closed simplicial 1-forms)

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Discrete Hodge for closed meshes

(adjoint complex with respect to Lsup2 inner products)

Discrete Hodge for closed meshes

Hodge decomposition is simple linear algebra (since dim is finite)(B Eckmannrsquo44 ldquoHarmonische Funktionenhelliprdquo)

Discrete Hodge for closed meshes

harmonic vector fieldsDiscrete Hodge decomposition

Discrete Hodge for closed meshes

harmonic vector fieldsDiscrete Hodge decomposition

Lemma Decomposition only depends on choice of spaces and inner product on It does neither

depend on inner products on and nor on choice of operators

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Discrete Harmonic vector fields

dim = 2genus

Discrete Harmonic vector fields

Solve sparse systemfor u and v

dim = 2genus

Get method for computing

Computing harmonic fields

dim = 2genus

1 Init H = 0 2 Pick random PC vector field X3 Compute potential and co-potential of X4 Compute harmonic part h of X5 h in H IF TRUE goto 2 ELSE H = span(Hh)6 dim(H) = 2genus IF TRUE exit ELSE goto 2

Complexity (4genus) sparse systems (because odds that step 5 returns TRUE are zero)

Las Vegas algorithm for computing harmonic fields

Computing harmonic fields

Harmonic fields cont

Harmonic fields cont

no tangential jump

X is closed

Harmonic fields cont

no tangential jump

X is co-closed

X is closed

Harmonic fields amp harmonic1-forms

(no tangential jump)

X is closed ie X corresponds tosimplicial 1-form

X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

this 1-form is co-closed(condition at vertices)X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

this 1-form is closed(condition at faces)X is pcw cnst

this 1-form is co-closed(condition at vertices)X is co-closed

this 1-form is co-closed(condition at vertices)

this 1-form is closed(condition at faces)X is pcw cnst harmonic 1-froms on graphs

cf GortlerGotsmanThurston

X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

(watch out cotan edge weightscould be negative cf BobenkoSpringborn forgetting non-negative weights by intrinsic edge flips)

Discrete Hodge Star

Complex multiplication (rotation by )

Discrete Hodge Star

X has no normal jump across edges

Complex multiplication (rotation by )

hellipdoes not take to itself

Because

X has no tangential jump across edges

X would have to be globally continuous

Discrete Hodge Star

Complex multiplication (rotation by )

Hodge star

hellipin other words Hodge star of X is harmonic part of J(X)

hellipdoes not take to itself

Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

(image are piecewise linear forms)

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

Adjoint operator

yields Whitney-Hodge decomposition

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57) by construction

deRham map DodziukPatodirsquo76

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of area

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Convergence of Hodge decomposition

Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)

bull This remains valid for variable (and converging) metrics

bull FEM-Whitney relations

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Thank you

• Algebraic Topology on Polyhedrafrom Linear Finite Elements
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Harmonic vector fields
• Contents
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Contents
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Contents
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Contents
• Discrete Harmonic vector fields
• Discrete Harmonic vector fields
• Computing harmonic fields
• Computing harmonic fields
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Discrete Hodge Star
• Discrete Hodge Star
• Discrete Hodge Star
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• Contents
• Convergence What can go wrong
• Convergence What can go wrong
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition

Letrsquos start with a concrete problem

Problem Compute space of harmonic vector fieldsfor meshes of genus gt 0

locally integrable byharmonic fcts

Applications Mesh parameterization (cf GuYau) texture mapping shape matching shape morphing hellip

Harmonic vector fields

Strategybull discretize deRham complexbull get model for cohomology Hsup1bull discretize Hodge decompositionbull get harmonic vector fieldsbull compute basis of harmonic vector fields

Problem Compute space of harmonic vector fieldsfor meshes of genus gt 0

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Review smooth deRham complex

functions 1-forms 2-forms

(Mg) smooth surface Its deRham complex is

d is Cartan outer differential (no metric)

Review smooth deRham complex

functions 1-forms 2-forms

(Mg) smooth surface Its deRham complex is

This is a chain complex

d is Cartan outer differential (no metric)

Review smooth deRham complex

functions 1-forms 2-forms

vector fields

(Mg) smooth surface Make use of metric g

Review smooth deRham complex

functions 1-forms 2-forms

(Mg) smooth surface Make use of metric g

vector fields

sharp Hodge star

Review smooth deRham complex

vector fields

functions 1-forms 2-forms

(Mg) smooth surface Make use of metric g

Review deRham cohomology

First deRham comology of smooth surface (Mg)from grad-curl chain complex

Review deRham cohomology

encodes global topology

First deRham comology of smooth surface (Mg)from grad-curl chain complex

Local integrability iff locally

Review deRham cohomology

First deRham comology of smooth surface (Mg)from grad-curl chain complex

encodes global topology

Review deRham cohomology

First deRham comology of smooth surface (Mg)from grad-curl chain complex

Strategy for discretization

Local integrability iff locally

encodes global topology

1st step2nd step

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Discrete deRham

space of continuous PL functions on polyhedron

Discrete gradient

Discrete deRham

space of continuous PL functions on polyhedron

Discrete gradient

bull Gradients are piecewise constant

Discrete deRham

space of continuous PL functions on polyhedron

Discrete gradient

bull Gradients are piecewise constantbull tangential components are equal

along any edge

Discrete deRham

space of continuous PL functions on polyhedron

bull Gradients are piecewise constantbull tangential components are equal

along any edge

Local integrability iff locally

Discrete gradient

Discrete deRham

Discrete curl

(tangential jump)

ifftangential components are equalalong any edge

Discrete deRham

(tangential jump)

PL functionson vertices

PL functionson edge midpoints

PC vectorfields

Discrete deRham complex

ifftangential components are equalalong any edge

Discrete deRham

Lagrangeelement

Crouzeix-Raviartrsquo73

Discrete deRham

Lagrangeelement

Crouzeix-Raviartrsquo73

If then complex is exact in middle comp

bull ArnoldFalkrsquo89 ldquoA uniformly accurate FEM for Reissner-Mindlin Platerdquobull PolthierPreussrsquo03 ldquoIdentifying Vector Field Singularitieshelliprdquo

Discrete deRham

In fact complex yields correct cohomology

(dim = 2genus)

(not surprising - think simplicial = closed simplicial 1-forms)

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Discrete Hodge for closed meshes

(adjoint complex with respect to Lsup2 inner products)

Discrete Hodge for closed meshes

Hodge decomposition is simple linear algebra (since dim is finite)(B Eckmannrsquo44 ldquoHarmonische Funktionenhelliprdquo)

Discrete Hodge for closed meshes

harmonic vector fieldsDiscrete Hodge decomposition

Discrete Hodge for closed meshes

harmonic vector fieldsDiscrete Hodge decomposition

Lemma Decomposition only depends on choice of spaces and inner product on It does neither

depend on inner products on and nor on choice of operators

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Discrete Harmonic vector fields

dim = 2genus

Discrete Harmonic vector fields

Solve sparse systemfor u and v

dim = 2genus

Get method for computing

Computing harmonic fields

dim = 2genus

1 Init H = 0 2 Pick random PC vector field X3 Compute potential and co-potential of X4 Compute harmonic part h of X5 h in H IF TRUE goto 2 ELSE H = span(Hh)6 dim(H) = 2genus IF TRUE exit ELSE goto 2

Complexity (4genus) sparse systems (because odds that step 5 returns TRUE are zero)

Las Vegas algorithm for computing harmonic fields

Computing harmonic fields

Harmonic fields cont

Harmonic fields cont

no tangential jump

X is closed

Harmonic fields cont

no tangential jump

X is co-closed

X is closed

Harmonic fields amp harmonic1-forms

(no tangential jump)

X is closed ie X corresponds tosimplicial 1-form

X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

this 1-form is co-closed(condition at vertices)X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

this 1-form is closed(condition at faces)X is pcw cnst

this 1-form is co-closed(condition at vertices)X is co-closed

this 1-form is co-closed(condition at vertices)

this 1-form is closed(condition at faces)X is pcw cnst harmonic 1-froms on graphs

cf GortlerGotsmanThurston

X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

(watch out cotan edge weightscould be negative cf BobenkoSpringborn forgetting non-negative weights by intrinsic edge flips)

Discrete Hodge Star

Complex multiplication (rotation by )

Discrete Hodge Star

X has no normal jump across edges

Complex multiplication (rotation by )

hellipdoes not take to itself

Because

X has no tangential jump across edges

X would have to be globally continuous

Discrete Hodge Star

Complex multiplication (rotation by )

Hodge star

hellipin other words Hodge star of X is harmonic part of J(X)

hellipdoes not take to itself

Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

(image are piecewise linear forms)

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

Adjoint operator

yields Whitney-Hodge decomposition

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57) by construction

deRham map DodziukPatodirsquo76

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of area

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Convergence of Hodge decomposition

Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)

bull This remains valid for variable (and converging) metrics

bull FEM-Whitney relations

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Thank you

• Algebraic Topology on Polyhedrafrom Linear Finite Elements
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Harmonic vector fields
• Contents
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Contents
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Contents
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Contents
• Discrete Harmonic vector fields
• Discrete Harmonic vector fields
• Computing harmonic fields
• Computing harmonic fields
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Discrete Hodge Star
• Discrete Hodge Star
• Discrete Hodge Star
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• Contents
• Convergence What can go wrong
• Convergence What can go wrong
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition

Harmonic vector fields

Strategybull discretize deRham complexbull get model for cohomology Hsup1bull discretize Hodge decompositionbull get harmonic vector fieldsbull compute basis of harmonic vector fields

Problem Compute space of harmonic vector fieldsfor meshes of genus gt 0

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Review smooth deRham complex

functions 1-forms 2-forms

(Mg) smooth surface Its deRham complex is

d is Cartan outer differential (no metric)

Review smooth deRham complex

functions 1-forms 2-forms

(Mg) smooth surface Its deRham complex is

This is a chain complex

d is Cartan outer differential (no metric)

Review smooth deRham complex

functions 1-forms 2-forms

vector fields

(Mg) smooth surface Make use of metric g

Review smooth deRham complex

functions 1-forms 2-forms

(Mg) smooth surface Make use of metric g

vector fields

sharp Hodge star

Review smooth deRham complex

vector fields

functions 1-forms 2-forms

(Mg) smooth surface Make use of metric g

Review deRham cohomology

First deRham comology of smooth surface (Mg)from grad-curl chain complex

Review deRham cohomology

encodes global topology

First deRham comology of smooth surface (Mg)from grad-curl chain complex

Local integrability iff locally

Review deRham cohomology

First deRham comology of smooth surface (Mg)from grad-curl chain complex

encodes global topology

Review deRham cohomology

First deRham comology of smooth surface (Mg)from grad-curl chain complex

Strategy for discretization

Local integrability iff locally

encodes global topology

1st step2nd step

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Discrete deRham

space of continuous PL functions on polyhedron

Discrete gradient

Discrete deRham

space of continuous PL functions on polyhedron

Discrete gradient

bull Gradients are piecewise constant

Discrete deRham

space of continuous PL functions on polyhedron

Discrete gradient

bull Gradients are piecewise constantbull tangential components are equal

along any edge

Discrete deRham

space of continuous PL functions on polyhedron

bull Gradients are piecewise constantbull tangential components are equal

along any edge

Local integrability iff locally

Discrete gradient

Discrete deRham

Discrete curl

(tangential jump)

ifftangential components are equalalong any edge

Discrete deRham

(tangential jump)

PL functionson vertices

PL functionson edge midpoints

PC vectorfields

Discrete deRham complex

ifftangential components are equalalong any edge

Discrete deRham

Lagrangeelement

Crouzeix-Raviartrsquo73

Discrete deRham

Lagrangeelement

Crouzeix-Raviartrsquo73

If then complex is exact in middle comp

bull ArnoldFalkrsquo89 ldquoA uniformly accurate FEM for Reissner-Mindlin Platerdquobull PolthierPreussrsquo03 ldquoIdentifying Vector Field Singularitieshelliprdquo

Discrete deRham

In fact complex yields correct cohomology

(dim = 2genus)

(not surprising - think simplicial = closed simplicial 1-forms)

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Discrete Hodge for closed meshes

(adjoint complex with respect to Lsup2 inner products)

Discrete Hodge for closed meshes

Hodge decomposition is simple linear algebra (since dim is finite)(B Eckmannrsquo44 ldquoHarmonische Funktionenhelliprdquo)

Discrete Hodge for closed meshes

harmonic vector fieldsDiscrete Hodge decomposition

Discrete Hodge for closed meshes

harmonic vector fieldsDiscrete Hodge decomposition

Lemma Decomposition only depends on choice of spaces and inner product on It does neither

depend on inner products on and nor on choice of operators

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Discrete Harmonic vector fields

dim = 2genus

Discrete Harmonic vector fields

Solve sparse systemfor u and v

dim = 2genus

Get method for computing

Computing harmonic fields

dim = 2genus

1 Init H = 0 2 Pick random PC vector field X3 Compute potential and co-potential of X4 Compute harmonic part h of X5 h in H IF TRUE goto 2 ELSE H = span(Hh)6 dim(H) = 2genus IF TRUE exit ELSE goto 2

Complexity (4genus) sparse systems (because odds that step 5 returns TRUE are zero)

Las Vegas algorithm for computing harmonic fields

Computing harmonic fields

Harmonic fields cont

Harmonic fields cont

no tangential jump

X is closed

Harmonic fields cont

no tangential jump

X is co-closed

X is closed

Harmonic fields amp harmonic1-forms

(no tangential jump)

X is closed ie X corresponds tosimplicial 1-form

X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

this 1-form is co-closed(condition at vertices)X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

this 1-form is closed(condition at faces)X is pcw cnst

this 1-form is co-closed(condition at vertices)X is co-closed

this 1-form is co-closed(condition at vertices)

this 1-form is closed(condition at faces)X is pcw cnst harmonic 1-froms on graphs

cf GortlerGotsmanThurston

X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

(watch out cotan edge weightscould be negative cf BobenkoSpringborn forgetting non-negative weights by intrinsic edge flips)

Discrete Hodge Star

Complex multiplication (rotation by )

Discrete Hodge Star

X has no normal jump across edges

Complex multiplication (rotation by )

hellipdoes not take to itself

Because

X has no tangential jump across edges

X would have to be globally continuous

Discrete Hodge Star

Complex multiplication (rotation by )

Hodge star

hellipin other words Hodge star of X is harmonic part of J(X)

hellipdoes not take to itself

Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

(image are piecewise linear forms)

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

Adjoint operator

yields Whitney-Hodge decomposition

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57) by construction

deRham map DodziukPatodirsquo76

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of area

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Convergence of Hodge decomposition

Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)

bull This remains valid for variable (and converging) metrics

bull FEM-Whitney relations

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Thank you

• Algebraic Topology on Polyhedrafrom Linear Finite Elements
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Harmonic vector fields
• Contents
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Contents
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Contents
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Contents
• Discrete Harmonic vector fields
• Discrete Harmonic vector fields
• Computing harmonic fields
• Computing harmonic fields
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Discrete Hodge Star
• Discrete Hodge Star
• Discrete Hodge Star
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• Contents
• Convergence What can go wrong
• Convergence What can go wrong
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Review smooth deRham complex

functions 1-forms 2-forms

(Mg) smooth surface Its deRham complex is

d is Cartan outer differential (no metric)

Review smooth deRham complex

functions 1-forms 2-forms

(Mg) smooth surface Its deRham complex is

This is a chain complex

d is Cartan outer differential (no metric)

Review smooth deRham complex

functions 1-forms 2-forms

vector fields

(Mg) smooth surface Make use of metric g

Review smooth deRham complex

functions 1-forms 2-forms

(Mg) smooth surface Make use of metric g

vector fields

sharp Hodge star

Review smooth deRham complex

vector fields

functions 1-forms 2-forms

(Mg) smooth surface Make use of metric g

Review deRham cohomology

First deRham comology of smooth surface (Mg)from grad-curl chain complex

Review deRham cohomology

encodes global topology

First deRham comology of smooth surface (Mg)from grad-curl chain complex

Local integrability iff locally

Review deRham cohomology

First deRham comology of smooth surface (Mg)from grad-curl chain complex

encodes global topology

Review deRham cohomology

First deRham comology of smooth surface (Mg)from grad-curl chain complex

Strategy for discretization

Local integrability iff locally

encodes global topology

1st step2nd step

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Discrete deRham

space of continuous PL functions on polyhedron

Discrete gradient

Discrete deRham

space of continuous PL functions on polyhedron

Discrete gradient

bull Gradients are piecewise constant

Discrete deRham

space of continuous PL functions on polyhedron

Discrete gradient

bull Gradients are piecewise constantbull tangential components are equal

along any edge

Discrete deRham

space of continuous PL functions on polyhedron

bull Gradients are piecewise constantbull tangential components are equal

along any edge

Local integrability iff locally

Discrete gradient

Discrete deRham

Discrete curl

(tangential jump)

ifftangential components are equalalong any edge

Discrete deRham

(tangential jump)

PL functionson vertices

PL functionson edge midpoints

PC vectorfields

Discrete deRham complex

ifftangential components are equalalong any edge

Discrete deRham

Lagrangeelement

Crouzeix-Raviartrsquo73

Discrete deRham

Lagrangeelement

Crouzeix-Raviartrsquo73

If then complex is exact in middle comp

bull ArnoldFalkrsquo89 ldquoA uniformly accurate FEM for Reissner-Mindlin Platerdquobull PolthierPreussrsquo03 ldquoIdentifying Vector Field Singularitieshelliprdquo

Discrete deRham

In fact complex yields correct cohomology

(dim = 2genus)

(not surprising - think simplicial = closed simplicial 1-forms)

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Discrete Hodge for closed meshes

(adjoint complex with respect to Lsup2 inner products)

Discrete Hodge for closed meshes

Hodge decomposition is simple linear algebra (since dim is finite)(B Eckmannrsquo44 ldquoHarmonische Funktionenhelliprdquo)

Discrete Hodge for closed meshes

harmonic vector fieldsDiscrete Hodge decomposition

Discrete Hodge for closed meshes

harmonic vector fieldsDiscrete Hodge decomposition

Lemma Decomposition only depends on choice of spaces and inner product on It does neither

depend on inner products on and nor on choice of operators

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Discrete Harmonic vector fields

dim = 2genus

Discrete Harmonic vector fields

Solve sparse systemfor u and v

dim = 2genus

Get method for computing

Computing harmonic fields

dim = 2genus

1 Init H = 0 2 Pick random PC vector field X3 Compute potential and co-potential of X4 Compute harmonic part h of X5 h in H IF TRUE goto 2 ELSE H = span(Hh)6 dim(H) = 2genus IF TRUE exit ELSE goto 2

Complexity (4genus) sparse systems (because odds that step 5 returns TRUE are zero)

Las Vegas algorithm for computing harmonic fields

Computing harmonic fields

Harmonic fields cont

Harmonic fields cont

no tangential jump

X is closed

Harmonic fields cont

no tangential jump

X is co-closed

X is closed

Harmonic fields amp harmonic1-forms

(no tangential jump)

X is closed ie X corresponds tosimplicial 1-form

X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

this 1-form is co-closed(condition at vertices)X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

this 1-form is closed(condition at faces)X is pcw cnst

this 1-form is co-closed(condition at vertices)X is co-closed

this 1-form is co-closed(condition at vertices)

this 1-form is closed(condition at faces)X is pcw cnst harmonic 1-froms on graphs

cf GortlerGotsmanThurston

X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

(watch out cotan edge weightscould be negative cf BobenkoSpringborn forgetting non-negative weights by intrinsic edge flips)

Discrete Hodge Star

Complex multiplication (rotation by )

Discrete Hodge Star

X has no normal jump across edges

Complex multiplication (rotation by )

hellipdoes not take to itself

Because

X has no tangential jump across edges

X would have to be globally continuous

Discrete Hodge Star

Complex multiplication (rotation by )

Hodge star

hellipin other words Hodge star of X is harmonic part of J(X)

hellipdoes not take to itself

Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

(image are piecewise linear forms)

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

Adjoint operator

yields Whitney-Hodge decomposition

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57) by construction

deRham map DodziukPatodirsquo76

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of area

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Convergence of Hodge decomposition

Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)

bull This remains valid for variable (and converging) metrics

bull FEM-Whitney relations

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Thank you

• Algebraic Topology on Polyhedrafrom Linear Finite Elements
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Harmonic vector fields
• Contents
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Contents
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Contents
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Contents
• Discrete Harmonic vector fields
• Discrete Harmonic vector fields
• Computing harmonic fields
• Computing harmonic fields
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Discrete Hodge Star
• Discrete Hodge Star
• Discrete Hodge Star
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• Contents
• Convergence What can go wrong
• Convergence What can go wrong
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition

Review smooth deRham complex

functions 1-forms 2-forms

(Mg) smooth surface Its deRham complex is

d is Cartan outer differential (no metric)

Review smooth deRham complex

functions 1-forms 2-forms

(Mg) smooth surface Its deRham complex is

This is a chain complex

d is Cartan outer differential (no metric)

Review smooth deRham complex

functions 1-forms 2-forms

vector fields

(Mg) smooth surface Make use of metric g

Review smooth deRham complex

functions 1-forms 2-forms

(Mg) smooth surface Make use of metric g

vector fields

sharp Hodge star

Review smooth deRham complex

vector fields

functions 1-forms 2-forms

(Mg) smooth surface Make use of metric g

Review deRham cohomology

First deRham comology of smooth surface (Mg)from grad-curl chain complex

Review deRham cohomology

encodes global topology

First deRham comology of smooth surface (Mg)from grad-curl chain complex

Local integrability iff locally

Review deRham cohomology

First deRham comology of smooth surface (Mg)from grad-curl chain complex

encodes global topology

Review deRham cohomology

First deRham comology of smooth surface (Mg)from grad-curl chain complex

Strategy for discretization

Local integrability iff locally

encodes global topology

1st step2nd step

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Discrete deRham

space of continuous PL functions on polyhedron

Discrete gradient

Discrete deRham

space of continuous PL functions on polyhedron

Discrete gradient

bull Gradients are piecewise constant

Discrete deRham

space of continuous PL functions on polyhedron

Discrete gradient

bull Gradients are piecewise constantbull tangential components are equal

along any edge

Discrete deRham

space of continuous PL functions on polyhedron

bull Gradients are piecewise constantbull tangential components are equal

along any edge

Local integrability iff locally

Discrete gradient

Discrete deRham

Discrete curl

(tangential jump)

ifftangential components are equalalong any edge

Discrete deRham

(tangential jump)

PL functionson vertices

PL functionson edge midpoints

PC vectorfields

Discrete deRham complex

ifftangential components are equalalong any edge

Discrete deRham

Lagrangeelement

Crouzeix-Raviartrsquo73

Discrete deRham

Lagrangeelement

Crouzeix-Raviartrsquo73

If then complex is exact in middle comp

bull ArnoldFalkrsquo89 ldquoA uniformly accurate FEM for Reissner-Mindlin Platerdquobull PolthierPreussrsquo03 ldquoIdentifying Vector Field Singularitieshelliprdquo

Discrete deRham

In fact complex yields correct cohomology

(dim = 2genus)

(not surprising - think simplicial = closed simplicial 1-forms)

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Discrete Hodge for closed meshes

(adjoint complex with respect to Lsup2 inner products)

Discrete Hodge for closed meshes

Hodge decomposition is simple linear algebra (since dim is finite)(B Eckmannrsquo44 ldquoHarmonische Funktionenhelliprdquo)

Discrete Hodge for closed meshes

harmonic vector fieldsDiscrete Hodge decomposition

Discrete Hodge for closed meshes

harmonic vector fieldsDiscrete Hodge decomposition

Lemma Decomposition only depends on choice of spaces and inner product on It does neither

depend on inner products on and nor on choice of operators

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Discrete Harmonic vector fields

dim = 2genus

Discrete Harmonic vector fields

Solve sparse systemfor u and v

dim = 2genus

Get method for computing

Computing harmonic fields

dim = 2genus

1 Init H = 0 2 Pick random PC vector field X3 Compute potential and co-potential of X4 Compute harmonic part h of X5 h in H IF TRUE goto 2 ELSE H = span(Hh)6 dim(H) = 2genus IF TRUE exit ELSE goto 2

Complexity (4genus) sparse systems (because odds that step 5 returns TRUE are zero)

Las Vegas algorithm for computing harmonic fields

Computing harmonic fields

Harmonic fields cont

Harmonic fields cont

no tangential jump

X is closed

Harmonic fields cont

no tangential jump

X is co-closed

X is closed

Harmonic fields amp harmonic1-forms

(no tangential jump)

X is closed ie X corresponds tosimplicial 1-form

X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

this 1-form is co-closed(condition at vertices)X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

this 1-form is closed(condition at faces)X is pcw cnst

this 1-form is co-closed(condition at vertices)X is co-closed

this 1-form is co-closed(condition at vertices)

this 1-form is closed(condition at faces)X is pcw cnst harmonic 1-froms on graphs

cf GortlerGotsmanThurston

X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

(watch out cotan edge weightscould be negative cf BobenkoSpringborn forgetting non-negative weights by intrinsic edge flips)

Discrete Hodge Star

Complex multiplication (rotation by )

Discrete Hodge Star

X has no normal jump across edges

Complex multiplication (rotation by )

hellipdoes not take to itself

Because

X has no tangential jump across edges

X would have to be globally continuous

Discrete Hodge Star

Complex multiplication (rotation by )

Hodge star

hellipin other words Hodge star of X is harmonic part of J(X)

hellipdoes not take to itself

Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

(image are piecewise linear forms)

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

Adjoint operator

yields Whitney-Hodge decomposition

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57) by construction

deRham map DodziukPatodirsquo76

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of area

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Convergence of Hodge decomposition

Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)

bull This remains valid for variable (and converging) metrics

bull FEM-Whitney relations

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Thank you

• Algebraic Topology on Polyhedrafrom Linear Finite Elements
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Harmonic vector fields
• Contents
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Contents
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Contents
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Contents
• Discrete Harmonic vector fields
• Discrete Harmonic vector fields
• Computing harmonic fields
• Computing harmonic fields
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Discrete Hodge Star
• Discrete Hodge Star
• Discrete Hodge Star
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• Contents
• Convergence What can go wrong
• Convergence What can go wrong
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition

Review smooth deRham complex

functions 1-forms 2-forms

(Mg) smooth surface Its deRham complex is

This is a chain complex

d is Cartan outer differential (no metric)

Review smooth deRham complex

functions 1-forms 2-forms

vector fields

(Mg) smooth surface Make use of metric g

Review smooth deRham complex

functions 1-forms 2-forms

(Mg) smooth surface Make use of metric g

vector fields

sharp Hodge star

Review smooth deRham complex

vector fields

functions 1-forms 2-forms

(Mg) smooth surface Make use of metric g

Review deRham cohomology

First deRham comology of smooth surface (Mg)from grad-curl chain complex

Review deRham cohomology

encodes global topology

First deRham comology of smooth surface (Mg)from grad-curl chain complex

Local integrability iff locally

Review deRham cohomology

First deRham comology of smooth surface (Mg)from grad-curl chain complex

encodes global topology

Review deRham cohomology

First deRham comology of smooth surface (Mg)from grad-curl chain complex

Strategy for discretization

Local integrability iff locally

encodes global topology

1st step2nd step

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Discrete deRham

space of continuous PL functions on polyhedron

Discrete gradient

Discrete deRham

space of continuous PL functions on polyhedron

Discrete gradient

bull Gradients are piecewise constant

Discrete deRham

space of continuous PL functions on polyhedron

Discrete gradient

bull Gradients are piecewise constantbull tangential components are equal

along any edge

Discrete deRham

space of continuous PL functions on polyhedron

bull Gradients are piecewise constantbull tangential components are equal

along any edge

Local integrability iff locally

Discrete gradient

Discrete deRham

Discrete curl

(tangential jump)

ifftangential components are equalalong any edge

Discrete deRham

(tangential jump)

PL functionson vertices

PL functionson edge midpoints

PC vectorfields

Discrete deRham complex

ifftangential components are equalalong any edge

Discrete deRham

Lagrangeelement

Crouzeix-Raviartrsquo73

Discrete deRham

Lagrangeelement

Crouzeix-Raviartrsquo73

If then complex is exact in middle comp

bull ArnoldFalkrsquo89 ldquoA uniformly accurate FEM for Reissner-Mindlin Platerdquobull PolthierPreussrsquo03 ldquoIdentifying Vector Field Singularitieshelliprdquo

Discrete deRham

In fact complex yields correct cohomology

(dim = 2genus)

(not surprising - think simplicial = closed simplicial 1-forms)

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Discrete Hodge for closed meshes

(adjoint complex with respect to Lsup2 inner products)

Discrete Hodge for closed meshes

Hodge decomposition is simple linear algebra (since dim is finite)(B Eckmannrsquo44 ldquoHarmonische Funktionenhelliprdquo)

Discrete Hodge for closed meshes

harmonic vector fieldsDiscrete Hodge decomposition

Discrete Hodge for closed meshes

harmonic vector fieldsDiscrete Hodge decomposition

Lemma Decomposition only depends on choice of spaces and inner product on It does neither

depend on inner products on and nor on choice of operators

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Discrete Harmonic vector fields

dim = 2genus

Discrete Harmonic vector fields

Solve sparse systemfor u and v

dim = 2genus

Get method for computing

Computing harmonic fields

dim = 2genus

1 Init H = 0 2 Pick random PC vector field X3 Compute potential and co-potential of X4 Compute harmonic part h of X5 h in H IF TRUE goto 2 ELSE H = span(Hh)6 dim(H) = 2genus IF TRUE exit ELSE goto 2

Complexity (4genus) sparse systems (because odds that step 5 returns TRUE are zero)

Las Vegas algorithm for computing harmonic fields

Computing harmonic fields

Harmonic fields cont

Harmonic fields cont

no tangential jump

X is closed

Harmonic fields cont

no tangential jump

X is co-closed

X is closed

Harmonic fields amp harmonic1-forms

(no tangential jump)

X is closed ie X corresponds tosimplicial 1-form

X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

this 1-form is co-closed(condition at vertices)X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

this 1-form is closed(condition at faces)X is pcw cnst

this 1-form is co-closed(condition at vertices)X is co-closed

this 1-form is co-closed(condition at vertices)

this 1-form is closed(condition at faces)X is pcw cnst harmonic 1-froms on graphs

cf GortlerGotsmanThurston

X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

(watch out cotan edge weightscould be negative cf BobenkoSpringborn forgetting non-negative weights by intrinsic edge flips)

Discrete Hodge Star

Complex multiplication (rotation by )

Discrete Hodge Star

X has no normal jump across edges

Complex multiplication (rotation by )

hellipdoes not take to itself

Because

X has no tangential jump across edges

X would have to be globally continuous

Discrete Hodge Star

Complex multiplication (rotation by )

Hodge star

hellipin other words Hodge star of X is harmonic part of J(X)

hellipdoes not take to itself

Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

(image are piecewise linear forms)

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

Adjoint operator

yields Whitney-Hodge decomposition

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57) by construction

deRham map DodziukPatodirsquo76

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of area

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Convergence of Hodge decomposition

Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)

bull This remains valid for variable (and converging) metrics

bull FEM-Whitney relations

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Thank you

• Algebraic Topology on Polyhedrafrom Linear Finite Elements
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Harmonic vector fields
• Contents
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Contents
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Contents
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Contents
• Discrete Harmonic vector fields
• Discrete Harmonic vector fields
• Computing harmonic fields
• Computing harmonic fields
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Discrete Hodge Star
• Discrete Hodge Star
• Discrete Hodge Star
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• Contents
• Convergence What can go wrong
• Convergence What can go wrong
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition

Review smooth deRham complex

functions 1-forms 2-forms

vector fields

(Mg) smooth surface Make use of metric g

Review smooth deRham complex

functions 1-forms 2-forms

(Mg) smooth surface Make use of metric g

vector fields

sharp Hodge star

Review smooth deRham complex

vector fields

functions 1-forms 2-forms

(Mg) smooth surface Make use of metric g

Review deRham cohomology

First deRham comology of smooth surface (Mg)from grad-curl chain complex

Review deRham cohomology

encodes global topology

First deRham comology of smooth surface (Mg)from grad-curl chain complex

Local integrability iff locally

Review deRham cohomology

First deRham comology of smooth surface (Mg)from grad-curl chain complex

encodes global topology

Review deRham cohomology

First deRham comology of smooth surface (Mg)from grad-curl chain complex

Strategy for discretization

Local integrability iff locally

encodes global topology

1st step2nd step

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Discrete deRham

space of continuous PL functions on polyhedron

Discrete gradient

Discrete deRham

space of continuous PL functions on polyhedron

Discrete gradient

bull Gradients are piecewise constant

Discrete deRham

space of continuous PL functions on polyhedron

Discrete gradient

bull Gradients are piecewise constantbull tangential components are equal

along any edge

Discrete deRham

space of continuous PL functions on polyhedron

bull Gradients are piecewise constantbull tangential components are equal

along any edge

Local integrability iff locally

Discrete gradient

Discrete deRham

Discrete curl

(tangential jump)

ifftangential components are equalalong any edge

Discrete deRham

(tangential jump)

PL functionson vertices

PL functionson edge midpoints

PC vectorfields

Discrete deRham complex

ifftangential components are equalalong any edge

Discrete deRham

Lagrangeelement

Crouzeix-Raviartrsquo73

Discrete deRham

Lagrangeelement

Crouzeix-Raviartrsquo73

If then complex is exact in middle comp

bull ArnoldFalkrsquo89 ldquoA uniformly accurate FEM for Reissner-Mindlin Platerdquobull PolthierPreussrsquo03 ldquoIdentifying Vector Field Singularitieshelliprdquo

Discrete deRham

In fact complex yields correct cohomology

(dim = 2genus)

(not surprising - think simplicial = closed simplicial 1-forms)

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Discrete Hodge for closed meshes

(adjoint complex with respect to Lsup2 inner products)

Discrete Hodge for closed meshes

Hodge decomposition is simple linear algebra (since dim is finite)(B Eckmannrsquo44 ldquoHarmonische Funktionenhelliprdquo)

Discrete Hodge for closed meshes

harmonic vector fieldsDiscrete Hodge decomposition

Discrete Hodge for closed meshes

harmonic vector fieldsDiscrete Hodge decomposition

Lemma Decomposition only depends on choice of spaces and inner product on It does neither

depend on inner products on and nor on choice of operators

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Discrete Harmonic vector fields

dim = 2genus

Discrete Harmonic vector fields

Solve sparse systemfor u and v

dim = 2genus

Get method for computing

Computing harmonic fields

dim = 2genus

1 Init H = 0 2 Pick random PC vector field X3 Compute potential and co-potential of X4 Compute harmonic part h of X5 h in H IF TRUE goto 2 ELSE H = span(Hh)6 dim(H) = 2genus IF TRUE exit ELSE goto 2

Complexity (4genus) sparse systems (because odds that step 5 returns TRUE are zero)

Las Vegas algorithm for computing harmonic fields

Computing harmonic fields

Harmonic fields cont

Harmonic fields cont

no tangential jump

X is closed

Harmonic fields cont

no tangential jump

X is co-closed

X is closed

Harmonic fields amp harmonic1-forms

(no tangential jump)

X is closed ie X corresponds tosimplicial 1-form

X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

this 1-form is co-closed(condition at vertices)X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

this 1-form is closed(condition at faces)X is pcw cnst

this 1-form is co-closed(condition at vertices)X is co-closed

this 1-form is co-closed(condition at vertices)

this 1-form is closed(condition at faces)X is pcw cnst harmonic 1-froms on graphs

cf GortlerGotsmanThurston

X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

(watch out cotan edge weightscould be negative cf BobenkoSpringborn forgetting non-negative weights by intrinsic edge flips)

Discrete Hodge Star

Complex multiplication (rotation by )

Discrete Hodge Star

X has no normal jump across edges

Complex multiplication (rotation by )

hellipdoes not take to itself

Because

X has no tangential jump across edges

X would have to be globally continuous

Discrete Hodge Star

Complex multiplication (rotation by )

Hodge star

hellipin other words Hodge star of X is harmonic part of J(X)

hellipdoes not take to itself

Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

(image are piecewise linear forms)

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

Adjoint operator

yields Whitney-Hodge decomposition

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57) by construction

deRham map DodziukPatodirsquo76

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of area

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Convergence of Hodge decomposition

Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)

bull This remains valid for variable (and converging) metrics

bull FEM-Whitney relations

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Thank you

• Algebraic Topology on Polyhedrafrom Linear Finite Elements
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Harmonic vector fields
• Contents
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Contents
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Contents
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Contents
• Discrete Harmonic vector fields
• Discrete Harmonic vector fields
• Computing harmonic fields
• Computing harmonic fields
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Discrete Hodge Star
• Discrete Hodge Star
• Discrete Hodge Star
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• Contents
• Convergence What can go wrong
• Convergence What can go wrong
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition

Review smooth deRham complex

functions 1-forms 2-forms

(Mg) smooth surface Make use of metric g

vector fields

sharp Hodge star

Review smooth deRham complex

vector fields

functions 1-forms 2-forms

(Mg) smooth surface Make use of metric g

Review deRham cohomology

First deRham comology of smooth surface (Mg)from grad-curl chain complex

Review deRham cohomology

encodes global topology

First deRham comology of smooth surface (Mg)from grad-curl chain complex

Local integrability iff locally

Review deRham cohomology

First deRham comology of smooth surface (Mg)from grad-curl chain complex

encodes global topology

Review deRham cohomology

First deRham comology of smooth surface (Mg)from grad-curl chain complex

Strategy for discretization

Local integrability iff locally

encodes global topology

1st step2nd step

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Discrete deRham

space of continuous PL functions on polyhedron

Discrete gradient

Discrete deRham

space of continuous PL functions on polyhedron

Discrete gradient

bull Gradients are piecewise constant

Discrete deRham

space of continuous PL functions on polyhedron

Discrete gradient

bull Gradients are piecewise constantbull tangential components are equal

along any edge

Discrete deRham

space of continuous PL functions on polyhedron

bull Gradients are piecewise constantbull tangential components are equal

along any edge

Local integrability iff locally

Discrete gradient

Discrete deRham

Discrete curl

(tangential jump)

ifftangential components are equalalong any edge

Discrete deRham

(tangential jump)

PL functionson vertices

PL functionson edge midpoints

PC vectorfields

Discrete deRham complex

ifftangential components are equalalong any edge

Discrete deRham

Lagrangeelement

Crouzeix-Raviartrsquo73

Discrete deRham

Lagrangeelement

Crouzeix-Raviartrsquo73

If then complex is exact in middle comp

bull ArnoldFalkrsquo89 ldquoA uniformly accurate FEM for Reissner-Mindlin Platerdquobull PolthierPreussrsquo03 ldquoIdentifying Vector Field Singularitieshelliprdquo

Discrete deRham

In fact complex yields correct cohomology

(dim = 2genus)

(not surprising - think simplicial = closed simplicial 1-forms)

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Discrete Hodge for closed meshes

(adjoint complex with respect to Lsup2 inner products)

Discrete Hodge for closed meshes

Hodge decomposition is simple linear algebra (since dim is finite)(B Eckmannrsquo44 ldquoHarmonische Funktionenhelliprdquo)

Discrete Hodge for closed meshes

harmonic vector fieldsDiscrete Hodge decomposition

Discrete Hodge for closed meshes

harmonic vector fieldsDiscrete Hodge decomposition

Lemma Decomposition only depends on choice of spaces and inner product on It does neither

depend on inner products on and nor on choice of operators

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Discrete Harmonic vector fields

dim = 2genus

Discrete Harmonic vector fields

Solve sparse systemfor u and v

dim = 2genus

Get method for computing

Computing harmonic fields

dim = 2genus

1 Init H = 0 2 Pick random PC vector field X3 Compute potential and co-potential of X4 Compute harmonic part h of X5 h in H IF TRUE goto 2 ELSE H = span(Hh)6 dim(H) = 2genus IF TRUE exit ELSE goto 2

Complexity (4genus) sparse systems (because odds that step 5 returns TRUE are zero)

Las Vegas algorithm for computing harmonic fields

Computing harmonic fields

Harmonic fields cont

Harmonic fields cont

no tangential jump

X is closed

Harmonic fields cont

no tangential jump

X is co-closed

X is closed

Harmonic fields amp harmonic1-forms

(no tangential jump)

X is closed ie X corresponds tosimplicial 1-form

X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

this 1-form is co-closed(condition at vertices)X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

this 1-form is closed(condition at faces)X is pcw cnst

this 1-form is co-closed(condition at vertices)X is co-closed

this 1-form is co-closed(condition at vertices)

this 1-form is closed(condition at faces)X is pcw cnst harmonic 1-froms on graphs

cf GortlerGotsmanThurston

X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

(watch out cotan edge weightscould be negative cf BobenkoSpringborn forgetting non-negative weights by intrinsic edge flips)

Discrete Hodge Star

Complex multiplication (rotation by )

Discrete Hodge Star

X has no normal jump across edges

Complex multiplication (rotation by )

hellipdoes not take to itself

Because

X has no tangential jump across edges

X would have to be globally continuous

Discrete Hodge Star

Complex multiplication (rotation by )

Hodge star

hellipin other words Hodge star of X is harmonic part of J(X)

hellipdoes not take to itself

Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

(image are piecewise linear forms)

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

Adjoint operator

yields Whitney-Hodge decomposition

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57) by construction

deRham map DodziukPatodirsquo76

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of area

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Convergence of Hodge decomposition

Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)

bull This remains valid for variable (and converging) metrics

bull FEM-Whitney relations

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Thank you

• Algebraic Topology on Polyhedrafrom Linear Finite Elements
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Harmonic vector fields
• Contents
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Contents
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Contents
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Contents
• Discrete Harmonic vector fields
• Discrete Harmonic vector fields
• Computing harmonic fields
• Computing harmonic fields
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Discrete Hodge Star
• Discrete Hodge Star
• Discrete Hodge Star
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• Contents
• Convergence What can go wrong
• Convergence What can go wrong
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition

Review smooth deRham complex

vector fields

functions 1-forms 2-forms

(Mg) smooth surface Make use of metric g

Review deRham cohomology

First deRham comology of smooth surface (Mg)from grad-curl chain complex

Review deRham cohomology

encodes global topology

First deRham comology of smooth surface (Mg)from grad-curl chain complex

Local integrability iff locally

Review deRham cohomology

First deRham comology of smooth surface (Mg)from grad-curl chain complex

encodes global topology

Review deRham cohomology

First deRham comology of smooth surface (Mg)from grad-curl chain complex

Strategy for discretization

Local integrability iff locally

encodes global topology

1st step2nd step

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Discrete deRham

space of continuous PL functions on polyhedron

Discrete gradient

Discrete deRham

space of continuous PL functions on polyhedron

Discrete gradient

bull Gradients are piecewise constant

Discrete deRham

space of continuous PL functions on polyhedron

Discrete gradient

bull Gradients are piecewise constantbull tangential components are equal

along any edge

Discrete deRham

space of continuous PL functions on polyhedron

bull Gradients are piecewise constantbull tangential components are equal

along any edge

Local integrability iff locally

Discrete gradient

Discrete deRham

Discrete curl

(tangential jump)

ifftangential components are equalalong any edge

Discrete deRham

(tangential jump)

PL functionson vertices

PL functionson edge midpoints

PC vectorfields

Discrete deRham complex

ifftangential components are equalalong any edge

Discrete deRham

Lagrangeelement

Crouzeix-Raviartrsquo73

Discrete deRham

Lagrangeelement

Crouzeix-Raviartrsquo73

If then complex is exact in middle comp

bull ArnoldFalkrsquo89 ldquoA uniformly accurate FEM for Reissner-Mindlin Platerdquobull PolthierPreussrsquo03 ldquoIdentifying Vector Field Singularitieshelliprdquo

Discrete deRham

In fact complex yields correct cohomology

(dim = 2genus)

(not surprising - think simplicial = closed simplicial 1-forms)

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Discrete Hodge for closed meshes

(adjoint complex with respect to Lsup2 inner products)

Discrete Hodge for closed meshes

Hodge decomposition is simple linear algebra (since dim is finite)(B Eckmannrsquo44 ldquoHarmonische Funktionenhelliprdquo)

Discrete Hodge for closed meshes

harmonic vector fieldsDiscrete Hodge decomposition

Discrete Hodge for closed meshes

harmonic vector fieldsDiscrete Hodge decomposition

Lemma Decomposition only depends on choice of spaces and inner product on It does neither

depend on inner products on and nor on choice of operators

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Discrete Harmonic vector fields

dim = 2genus

Discrete Harmonic vector fields

Solve sparse systemfor u and v

dim = 2genus

Get method for computing

Computing harmonic fields

dim = 2genus

1 Init H = 0 2 Pick random PC vector field X3 Compute potential and co-potential of X4 Compute harmonic part h of X5 h in H IF TRUE goto 2 ELSE H = span(Hh)6 dim(H) = 2genus IF TRUE exit ELSE goto 2

Complexity (4genus) sparse systems (because odds that step 5 returns TRUE are zero)

Las Vegas algorithm for computing harmonic fields

Computing harmonic fields

Harmonic fields cont

Harmonic fields cont

no tangential jump

X is closed

Harmonic fields cont

no tangential jump

X is co-closed

X is closed

Harmonic fields amp harmonic1-forms

(no tangential jump)

X is closed ie X corresponds tosimplicial 1-form

X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

this 1-form is co-closed(condition at vertices)X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

this 1-form is closed(condition at faces)X is pcw cnst

this 1-form is co-closed(condition at vertices)X is co-closed

this 1-form is co-closed(condition at vertices)

this 1-form is closed(condition at faces)X is pcw cnst harmonic 1-froms on graphs

cf GortlerGotsmanThurston

X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

(watch out cotan edge weightscould be negative cf BobenkoSpringborn forgetting non-negative weights by intrinsic edge flips)

Discrete Hodge Star

Complex multiplication (rotation by )

Discrete Hodge Star

X has no normal jump across edges

Complex multiplication (rotation by )

hellipdoes not take to itself

Because

X has no tangential jump across edges

X would have to be globally continuous

Discrete Hodge Star

Complex multiplication (rotation by )

Hodge star

hellipin other words Hodge star of X is harmonic part of J(X)

hellipdoes not take to itself

Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

(image are piecewise linear forms)

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

Adjoint operator

yields Whitney-Hodge decomposition

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57) by construction

deRham map DodziukPatodirsquo76

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of area

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Convergence of Hodge decomposition

Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)

bull This remains valid for variable (and converging) metrics

bull FEM-Whitney relations

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Thank you

• Algebraic Topology on Polyhedrafrom Linear Finite Elements
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Harmonic vector fields
• Contents
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Contents
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Contents
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Contents
• Discrete Harmonic vector fields
• Discrete Harmonic vector fields
• Computing harmonic fields
• Computing harmonic fields
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Discrete Hodge Star
• Discrete Hodge Star
• Discrete Hodge Star
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• Contents
• Convergence What can go wrong
• Convergence What can go wrong
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition

Review deRham cohomology

First deRham comology of smooth surface (Mg)from grad-curl chain complex

Review deRham cohomology

encodes global topology

First deRham comology of smooth surface (Mg)from grad-curl chain complex

Local integrability iff locally

Review deRham cohomology

First deRham comology of smooth surface (Mg)from grad-curl chain complex

encodes global topology

Review deRham cohomology

First deRham comology of smooth surface (Mg)from grad-curl chain complex

Strategy for discretization

Local integrability iff locally

encodes global topology

1st step2nd step

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Discrete deRham

space of continuous PL functions on polyhedron

Discrete gradient

Discrete deRham

space of continuous PL functions on polyhedron

Discrete gradient

bull Gradients are piecewise constant

Discrete deRham

space of continuous PL functions on polyhedron

Discrete gradient

bull Gradients are piecewise constantbull tangential components are equal

along any edge

Discrete deRham

space of continuous PL functions on polyhedron

bull Gradients are piecewise constantbull tangential components are equal

along any edge

Local integrability iff locally

Discrete gradient

Discrete deRham

Discrete curl

(tangential jump)

ifftangential components are equalalong any edge

Discrete deRham

(tangential jump)

PL functionson vertices

PL functionson edge midpoints

PC vectorfields

Discrete deRham complex

ifftangential components are equalalong any edge

Discrete deRham

Lagrangeelement

Crouzeix-Raviartrsquo73

Discrete deRham

Lagrangeelement

Crouzeix-Raviartrsquo73

If then complex is exact in middle comp

bull ArnoldFalkrsquo89 ldquoA uniformly accurate FEM for Reissner-Mindlin Platerdquobull PolthierPreussrsquo03 ldquoIdentifying Vector Field Singularitieshelliprdquo

Discrete deRham

In fact complex yields correct cohomology

(dim = 2genus)

(not surprising - think simplicial = closed simplicial 1-forms)

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Discrete Hodge for closed meshes

(adjoint complex with respect to Lsup2 inner products)

Discrete Hodge for closed meshes

Hodge decomposition is simple linear algebra (since dim is finite)(B Eckmannrsquo44 ldquoHarmonische Funktionenhelliprdquo)

Discrete Hodge for closed meshes

harmonic vector fieldsDiscrete Hodge decomposition

Discrete Hodge for closed meshes

harmonic vector fieldsDiscrete Hodge decomposition

Lemma Decomposition only depends on choice of spaces and inner product on It does neither

depend on inner products on and nor on choice of operators

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Discrete Harmonic vector fields

dim = 2genus

Discrete Harmonic vector fields

Solve sparse systemfor u and v

dim = 2genus

Get method for computing

Computing harmonic fields

dim = 2genus

1 Init H = 0 2 Pick random PC vector field X3 Compute potential and co-potential of X4 Compute harmonic part h of X5 h in H IF TRUE goto 2 ELSE H = span(Hh)6 dim(H) = 2genus IF TRUE exit ELSE goto 2

Complexity (4genus) sparse systems (because odds that step 5 returns TRUE are zero)

Las Vegas algorithm for computing harmonic fields

Computing harmonic fields

Harmonic fields cont

Harmonic fields cont

no tangential jump

X is closed

Harmonic fields cont

no tangential jump

X is co-closed

X is closed

Harmonic fields amp harmonic1-forms

(no tangential jump)

X is closed ie X corresponds tosimplicial 1-form

X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

this 1-form is co-closed(condition at vertices)X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

this 1-form is closed(condition at faces)X is pcw cnst

this 1-form is co-closed(condition at vertices)X is co-closed

this 1-form is co-closed(condition at vertices)

this 1-form is closed(condition at faces)X is pcw cnst harmonic 1-froms on graphs

cf GortlerGotsmanThurston

X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

(watch out cotan edge weightscould be negative cf BobenkoSpringborn forgetting non-negative weights by intrinsic edge flips)

Discrete Hodge Star

Complex multiplication (rotation by )

Discrete Hodge Star

X has no normal jump across edges

Complex multiplication (rotation by )

hellipdoes not take to itself

Because

X has no tangential jump across edges

X would have to be globally continuous

Discrete Hodge Star

Complex multiplication (rotation by )

Hodge star

hellipin other words Hodge star of X is harmonic part of J(X)

hellipdoes not take to itself

Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

(image are piecewise linear forms)

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

Adjoint operator

yields Whitney-Hodge decomposition

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57) by construction

deRham map DodziukPatodirsquo76

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of area

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Convergence of Hodge decomposition

Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)

bull This remains valid for variable (and converging) metrics

bull FEM-Whitney relations

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Thank you

• Algebraic Topology on Polyhedrafrom Linear Finite Elements
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Harmonic vector fields
• Contents
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Contents
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Contents
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Contents
• Discrete Harmonic vector fields
• Discrete Harmonic vector fields
• Computing harmonic fields
• Computing harmonic fields
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Discrete Hodge Star
• Discrete Hodge Star
• Discrete Hodge Star
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• Contents
• Convergence What can go wrong
• Convergence What can go wrong
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition

Review deRham cohomology

encodes global topology

First deRham comology of smooth surface (Mg)from grad-curl chain complex

Local integrability iff locally

Review deRham cohomology

First deRham comology of smooth surface (Mg)from grad-curl chain complex

encodes global topology

Review deRham cohomology

First deRham comology of smooth surface (Mg)from grad-curl chain complex

Strategy for discretization

Local integrability iff locally

encodes global topology

1st step2nd step

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Discrete deRham

space of continuous PL functions on polyhedron

Discrete gradient

Discrete deRham

space of continuous PL functions on polyhedron

Discrete gradient

bull Gradients are piecewise constant

Discrete deRham

space of continuous PL functions on polyhedron

Discrete gradient

bull Gradients are piecewise constantbull tangential components are equal

along any edge

Discrete deRham

space of continuous PL functions on polyhedron

bull Gradients are piecewise constantbull tangential components are equal

along any edge

Local integrability iff locally

Discrete gradient

Discrete deRham

Discrete curl

(tangential jump)

ifftangential components are equalalong any edge

Discrete deRham

(tangential jump)

PL functionson vertices

PL functionson edge midpoints

PC vectorfields

Discrete deRham complex

ifftangential components are equalalong any edge

Discrete deRham

Lagrangeelement

Crouzeix-Raviartrsquo73

Discrete deRham

Lagrangeelement

Crouzeix-Raviartrsquo73

If then complex is exact in middle comp

bull ArnoldFalkrsquo89 ldquoA uniformly accurate FEM for Reissner-Mindlin Platerdquobull PolthierPreussrsquo03 ldquoIdentifying Vector Field Singularitieshelliprdquo

Discrete deRham

In fact complex yields correct cohomology

(dim = 2genus)

(not surprising - think simplicial = closed simplicial 1-forms)

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Discrete Hodge for closed meshes

(adjoint complex with respect to Lsup2 inner products)

Discrete Hodge for closed meshes

Hodge decomposition is simple linear algebra (since dim is finite)(B Eckmannrsquo44 ldquoHarmonische Funktionenhelliprdquo)

Discrete Hodge for closed meshes

harmonic vector fieldsDiscrete Hodge decomposition

Discrete Hodge for closed meshes

harmonic vector fieldsDiscrete Hodge decomposition

Lemma Decomposition only depends on choice of spaces and inner product on It does neither

depend on inner products on and nor on choice of operators

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Discrete Harmonic vector fields

dim = 2genus

Discrete Harmonic vector fields

Solve sparse systemfor u and v

dim = 2genus

Get method for computing

Computing harmonic fields

dim = 2genus

1 Init H = 0 2 Pick random PC vector field X3 Compute potential and co-potential of X4 Compute harmonic part h of X5 h in H IF TRUE goto 2 ELSE H = span(Hh)6 dim(H) = 2genus IF TRUE exit ELSE goto 2

Complexity (4genus) sparse systems (because odds that step 5 returns TRUE are zero)

Las Vegas algorithm for computing harmonic fields

Computing harmonic fields

Harmonic fields cont

Harmonic fields cont

no tangential jump

X is closed

Harmonic fields cont

no tangential jump

X is co-closed

X is closed

Harmonic fields amp harmonic1-forms

(no tangential jump)

X is closed ie X corresponds tosimplicial 1-form

X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

this 1-form is co-closed(condition at vertices)X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

this 1-form is closed(condition at faces)X is pcw cnst

this 1-form is co-closed(condition at vertices)X is co-closed

this 1-form is co-closed(condition at vertices)

this 1-form is closed(condition at faces)X is pcw cnst harmonic 1-froms on graphs

cf GortlerGotsmanThurston

X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

(watch out cotan edge weightscould be negative cf BobenkoSpringborn forgetting non-negative weights by intrinsic edge flips)

Discrete Hodge Star

Complex multiplication (rotation by )

Discrete Hodge Star

X has no normal jump across edges

Complex multiplication (rotation by )

hellipdoes not take to itself

Because

X has no tangential jump across edges

X would have to be globally continuous

Discrete Hodge Star

Complex multiplication (rotation by )

Hodge star

hellipin other words Hodge star of X is harmonic part of J(X)

hellipdoes not take to itself

Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

(image are piecewise linear forms)

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

Adjoint operator

yields Whitney-Hodge decomposition

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57) by construction

deRham map DodziukPatodirsquo76

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of area

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Convergence of Hodge decomposition

Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)

bull This remains valid for variable (and converging) metrics

bull FEM-Whitney relations

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Thank you

• Algebraic Topology on Polyhedrafrom Linear Finite Elements
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Harmonic vector fields
• Contents
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Contents
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Contents
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Contents
• Discrete Harmonic vector fields
• Discrete Harmonic vector fields
• Computing harmonic fields
• Computing harmonic fields
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Discrete Hodge Star
• Discrete Hodge Star
• Discrete Hodge Star
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• Contents
• Convergence What can go wrong
• Convergence What can go wrong
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition

Local integrability iff locally

Review deRham cohomology

First deRham comology of smooth surface (Mg)from grad-curl chain complex

encodes global topology

Review deRham cohomology

First deRham comology of smooth surface (Mg)from grad-curl chain complex

Strategy for discretization

Local integrability iff locally

encodes global topology

1st step2nd step

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Discrete deRham

space of continuous PL functions on polyhedron

Discrete gradient

Discrete deRham

space of continuous PL functions on polyhedron

Discrete gradient

bull Gradients are piecewise constant

Discrete deRham

space of continuous PL functions on polyhedron

Discrete gradient

bull Gradients are piecewise constantbull tangential components are equal

along any edge

Discrete deRham

space of continuous PL functions on polyhedron

bull Gradients are piecewise constantbull tangential components are equal

along any edge

Local integrability iff locally

Discrete gradient

Discrete deRham

Discrete curl

(tangential jump)

ifftangential components are equalalong any edge

Discrete deRham

(tangential jump)

PL functionson vertices

PL functionson edge midpoints

PC vectorfields

Discrete deRham complex

ifftangential components are equalalong any edge

Discrete deRham

Lagrangeelement

Crouzeix-Raviartrsquo73

Discrete deRham

Lagrangeelement

Crouzeix-Raviartrsquo73

If then complex is exact in middle comp

bull ArnoldFalkrsquo89 ldquoA uniformly accurate FEM for Reissner-Mindlin Platerdquobull PolthierPreussrsquo03 ldquoIdentifying Vector Field Singularitieshelliprdquo

Discrete deRham

In fact complex yields correct cohomology

(dim = 2genus)

(not surprising - think simplicial = closed simplicial 1-forms)

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Discrete Hodge for closed meshes

(adjoint complex with respect to Lsup2 inner products)

Discrete Hodge for closed meshes

Hodge decomposition is simple linear algebra (since dim is finite)(B Eckmannrsquo44 ldquoHarmonische Funktionenhelliprdquo)

Discrete Hodge for closed meshes

harmonic vector fieldsDiscrete Hodge decomposition

Discrete Hodge for closed meshes

harmonic vector fieldsDiscrete Hodge decomposition

Lemma Decomposition only depends on choice of spaces and inner product on It does neither

depend on inner products on and nor on choice of operators

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Discrete Harmonic vector fields

dim = 2genus

Discrete Harmonic vector fields

Solve sparse systemfor u and v

dim = 2genus

Get method for computing

Computing harmonic fields

dim = 2genus

1 Init H = 0 2 Pick random PC vector field X3 Compute potential and co-potential of X4 Compute harmonic part h of X5 h in H IF TRUE goto 2 ELSE H = span(Hh)6 dim(H) = 2genus IF TRUE exit ELSE goto 2

Complexity (4genus) sparse systems (because odds that step 5 returns TRUE are zero)

Las Vegas algorithm for computing harmonic fields

Computing harmonic fields

Harmonic fields cont

Harmonic fields cont

no tangential jump

X is closed

Harmonic fields cont

no tangential jump

X is co-closed

X is closed

Harmonic fields amp harmonic1-forms

(no tangential jump)

X is closed ie X corresponds tosimplicial 1-form

X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

this 1-form is co-closed(condition at vertices)X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

this 1-form is closed(condition at faces)X is pcw cnst

this 1-form is co-closed(condition at vertices)X is co-closed

this 1-form is co-closed(condition at vertices)

this 1-form is closed(condition at faces)X is pcw cnst harmonic 1-froms on graphs

cf GortlerGotsmanThurston

X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

(watch out cotan edge weightscould be negative cf BobenkoSpringborn forgetting non-negative weights by intrinsic edge flips)

Discrete Hodge Star

Complex multiplication (rotation by )

Discrete Hodge Star

X has no normal jump across edges

Complex multiplication (rotation by )

hellipdoes not take to itself

Because

X has no tangential jump across edges

X would have to be globally continuous

Discrete Hodge Star

Complex multiplication (rotation by )

Hodge star

hellipin other words Hodge star of X is harmonic part of J(X)

hellipdoes not take to itself

Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

(image are piecewise linear forms)

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

Adjoint operator

yields Whitney-Hodge decomposition

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57) by construction

deRham map DodziukPatodirsquo76

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of area

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Convergence of Hodge decomposition

Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)

bull This remains valid for variable (and converging) metrics

bull FEM-Whitney relations

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Thank you

• Algebraic Topology on Polyhedrafrom Linear Finite Elements
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Harmonic vector fields
• Contents
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Contents
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Contents
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Contents
• Discrete Harmonic vector fields
• Discrete Harmonic vector fields
• Computing harmonic fields
• Computing harmonic fields
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Discrete Hodge Star
• Discrete Hodge Star
• Discrete Hodge Star
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• Contents
• Convergence What can go wrong
• Convergence What can go wrong
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition

Review deRham cohomology

First deRham comology of smooth surface (Mg)from grad-curl chain complex

Strategy for discretization

Local integrability iff locally

encodes global topology

1st step2nd step

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Discrete deRham

space of continuous PL functions on polyhedron

Discrete gradient

Discrete deRham

space of continuous PL functions on polyhedron

Discrete gradient

bull Gradients are piecewise constant

Discrete deRham

space of continuous PL functions on polyhedron

Discrete gradient

bull Gradients are piecewise constantbull tangential components are equal

along any edge

Discrete deRham

space of continuous PL functions on polyhedron

bull Gradients are piecewise constantbull tangential components are equal

along any edge

Local integrability iff locally

Discrete gradient

Discrete deRham

Discrete curl

(tangential jump)

ifftangential components are equalalong any edge

Discrete deRham

(tangential jump)

PL functionson vertices

PL functionson edge midpoints

PC vectorfields

Discrete deRham complex

ifftangential components are equalalong any edge

Discrete deRham

Lagrangeelement

Crouzeix-Raviartrsquo73

Discrete deRham

Lagrangeelement

Crouzeix-Raviartrsquo73

If then complex is exact in middle comp

bull ArnoldFalkrsquo89 ldquoA uniformly accurate FEM for Reissner-Mindlin Platerdquobull PolthierPreussrsquo03 ldquoIdentifying Vector Field Singularitieshelliprdquo

Discrete deRham

In fact complex yields correct cohomology

(dim = 2genus)

(not surprising - think simplicial = closed simplicial 1-forms)

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Discrete Hodge for closed meshes

(adjoint complex with respect to Lsup2 inner products)

Discrete Hodge for closed meshes

Hodge decomposition is simple linear algebra (since dim is finite)(B Eckmannrsquo44 ldquoHarmonische Funktionenhelliprdquo)

Discrete Hodge for closed meshes

harmonic vector fieldsDiscrete Hodge decomposition

Discrete Hodge for closed meshes

harmonic vector fieldsDiscrete Hodge decomposition

Lemma Decomposition only depends on choice of spaces and inner product on It does neither

depend on inner products on and nor on choice of operators

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Discrete Harmonic vector fields

dim = 2genus

Discrete Harmonic vector fields

Solve sparse systemfor u and v

dim = 2genus

Get method for computing

Computing harmonic fields

dim = 2genus

1 Init H = 0 2 Pick random PC vector field X3 Compute potential and co-potential of X4 Compute harmonic part h of X5 h in H IF TRUE goto 2 ELSE H = span(Hh)6 dim(H) = 2genus IF TRUE exit ELSE goto 2

Complexity (4genus) sparse systems (because odds that step 5 returns TRUE are zero)

Las Vegas algorithm for computing harmonic fields

Computing harmonic fields

Harmonic fields cont

Harmonic fields cont

no tangential jump

X is closed

Harmonic fields cont

no tangential jump

X is co-closed

X is closed

Harmonic fields amp harmonic1-forms

(no tangential jump)

X is closed ie X corresponds tosimplicial 1-form

X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

this 1-form is co-closed(condition at vertices)X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

this 1-form is closed(condition at faces)X is pcw cnst

this 1-form is co-closed(condition at vertices)X is co-closed

this 1-form is co-closed(condition at vertices)

this 1-form is closed(condition at faces)X is pcw cnst harmonic 1-froms on graphs

cf GortlerGotsmanThurston

X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

(watch out cotan edge weightscould be negative cf BobenkoSpringborn forgetting non-negative weights by intrinsic edge flips)

Discrete Hodge Star

Complex multiplication (rotation by )

Discrete Hodge Star

X has no normal jump across edges

Complex multiplication (rotation by )

hellipdoes not take to itself

Because

X has no tangential jump across edges

X would have to be globally continuous

Discrete Hodge Star

Complex multiplication (rotation by )

Hodge star

hellipin other words Hodge star of X is harmonic part of J(X)

hellipdoes not take to itself

Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

(image are piecewise linear forms)

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

Adjoint operator

yields Whitney-Hodge decomposition

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57) by construction

deRham map DodziukPatodirsquo76

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of area

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Convergence of Hodge decomposition

Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)

bull This remains valid for variable (and converging) metrics

bull FEM-Whitney relations

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Thank you

• Algebraic Topology on Polyhedrafrom Linear Finite Elements
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Harmonic vector fields
• Contents
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Contents
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Contents
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Contents
• Discrete Harmonic vector fields
• Discrete Harmonic vector fields
• Computing harmonic fields
• Computing harmonic fields
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Discrete Hodge Star
• Discrete Hodge Star
• Discrete Hodge Star
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• Contents
• Convergence What can go wrong
• Convergence What can go wrong
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Discrete deRham

space of continuous PL functions on polyhedron

Discrete gradient

Discrete deRham

space of continuous PL functions on polyhedron

Discrete gradient

bull Gradients are piecewise constant

Discrete deRham

space of continuous PL functions on polyhedron

Discrete gradient

bull Gradients are piecewise constantbull tangential components are equal

along any edge

Discrete deRham

space of continuous PL functions on polyhedron

bull Gradients are piecewise constantbull tangential components are equal

along any edge

Local integrability iff locally

Discrete gradient

Discrete deRham

Discrete curl

(tangential jump)

ifftangential components are equalalong any edge

Discrete deRham

(tangential jump)

PL functionson vertices

PL functionson edge midpoints

PC vectorfields

Discrete deRham complex

ifftangential components are equalalong any edge

Discrete deRham

Lagrangeelement

Crouzeix-Raviartrsquo73

Discrete deRham

Lagrangeelement

Crouzeix-Raviartrsquo73

If then complex is exact in middle comp

bull ArnoldFalkrsquo89 ldquoA uniformly accurate FEM for Reissner-Mindlin Platerdquobull PolthierPreussrsquo03 ldquoIdentifying Vector Field Singularitieshelliprdquo

Discrete deRham

In fact complex yields correct cohomology

(dim = 2genus)

(not surprising - think simplicial = closed simplicial 1-forms)

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Discrete Hodge for closed meshes

(adjoint complex with respect to Lsup2 inner products)

Discrete Hodge for closed meshes

Hodge decomposition is simple linear algebra (since dim is finite)(B Eckmannrsquo44 ldquoHarmonische Funktionenhelliprdquo)

Discrete Hodge for closed meshes

harmonic vector fieldsDiscrete Hodge decomposition

Discrete Hodge for closed meshes

harmonic vector fieldsDiscrete Hodge decomposition

Lemma Decomposition only depends on choice of spaces and inner product on It does neither

depend on inner products on and nor on choice of operators

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Discrete Harmonic vector fields

dim = 2genus

Discrete Harmonic vector fields

Solve sparse systemfor u and v

dim = 2genus

Get method for computing

Computing harmonic fields

dim = 2genus

1 Init H = 0 2 Pick random PC vector field X3 Compute potential and co-potential of X4 Compute harmonic part h of X5 h in H IF TRUE goto 2 ELSE H = span(Hh)6 dim(H) = 2genus IF TRUE exit ELSE goto 2

Complexity (4genus) sparse systems (because odds that step 5 returns TRUE are zero)

Las Vegas algorithm for computing harmonic fields

Computing harmonic fields

Harmonic fields cont

Harmonic fields cont

no tangential jump

X is closed

Harmonic fields cont

no tangential jump

X is co-closed

X is closed

Harmonic fields amp harmonic1-forms

(no tangential jump)

X is closed ie X corresponds tosimplicial 1-form

X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

this 1-form is co-closed(condition at vertices)X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

this 1-form is closed(condition at faces)X is pcw cnst

this 1-form is co-closed(condition at vertices)X is co-closed

this 1-form is co-closed(condition at vertices)

this 1-form is closed(condition at faces)X is pcw cnst harmonic 1-froms on graphs

cf GortlerGotsmanThurston

X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

(watch out cotan edge weightscould be negative cf BobenkoSpringborn forgetting non-negative weights by intrinsic edge flips)

Discrete Hodge Star

Complex multiplication (rotation by )

Discrete Hodge Star

X has no normal jump across edges

Complex multiplication (rotation by )

hellipdoes not take to itself

Because

X has no tangential jump across edges

X would have to be globally continuous

Discrete Hodge Star

Complex multiplication (rotation by )

Hodge star

hellipin other words Hodge star of X is harmonic part of J(X)

hellipdoes not take to itself

Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

(image are piecewise linear forms)

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

Adjoint operator

yields Whitney-Hodge decomposition

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57) by construction

deRham map DodziukPatodirsquo76

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of area

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Convergence of Hodge decomposition

Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)

bull This remains valid for variable (and converging) metrics

bull FEM-Whitney relations

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Thank you

• Algebraic Topology on Polyhedrafrom Linear Finite Elements
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Harmonic vector fields
• Contents
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Contents
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Contents
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Contents
• Discrete Harmonic vector fields
• Discrete Harmonic vector fields
• Computing harmonic fields
• Computing harmonic fields
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Discrete Hodge Star
• Discrete Hodge Star
• Discrete Hodge Star
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• Contents
• Convergence What can go wrong
• Convergence What can go wrong
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition

Discrete deRham

space of continuous PL functions on polyhedron

Discrete gradient

Discrete deRham

space of continuous PL functions on polyhedron

Discrete gradient

bull Gradients are piecewise constant

Discrete deRham

space of continuous PL functions on polyhedron

Discrete gradient

bull Gradients are piecewise constantbull tangential components are equal

along any edge

Discrete deRham

space of continuous PL functions on polyhedron

bull Gradients are piecewise constantbull tangential components are equal

along any edge

Local integrability iff locally

Discrete gradient

Discrete deRham

Discrete curl

(tangential jump)

ifftangential components are equalalong any edge

Discrete deRham

(tangential jump)

PL functionson vertices

PL functionson edge midpoints

PC vectorfields

Discrete deRham complex

ifftangential components are equalalong any edge

Discrete deRham

Lagrangeelement

Crouzeix-Raviartrsquo73

Discrete deRham

Lagrangeelement

Crouzeix-Raviartrsquo73

If then complex is exact in middle comp

bull ArnoldFalkrsquo89 ldquoA uniformly accurate FEM for Reissner-Mindlin Platerdquobull PolthierPreussrsquo03 ldquoIdentifying Vector Field Singularitieshelliprdquo

Discrete deRham

In fact complex yields correct cohomology

(dim = 2genus)

(not surprising - think simplicial = closed simplicial 1-forms)

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Discrete Hodge for closed meshes

(adjoint complex with respect to Lsup2 inner products)

Discrete Hodge for closed meshes

Hodge decomposition is simple linear algebra (since dim is finite)(B Eckmannrsquo44 ldquoHarmonische Funktionenhelliprdquo)

Discrete Hodge for closed meshes

harmonic vector fieldsDiscrete Hodge decomposition

Discrete Hodge for closed meshes

harmonic vector fieldsDiscrete Hodge decomposition

Lemma Decomposition only depends on choice of spaces and inner product on It does neither

depend on inner products on and nor on choice of operators

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Discrete Harmonic vector fields

dim = 2genus

Discrete Harmonic vector fields

Solve sparse systemfor u and v

dim = 2genus

Get method for computing

Computing harmonic fields

dim = 2genus

1 Init H = 0 2 Pick random PC vector field X3 Compute potential and co-potential of X4 Compute harmonic part h of X5 h in H IF TRUE goto 2 ELSE H = span(Hh)6 dim(H) = 2genus IF TRUE exit ELSE goto 2

Complexity (4genus) sparse systems (because odds that step 5 returns TRUE are zero)

Las Vegas algorithm for computing harmonic fields

Computing harmonic fields

Harmonic fields cont

Harmonic fields cont

no tangential jump

X is closed

Harmonic fields cont

no tangential jump

X is co-closed

X is closed

Harmonic fields amp harmonic1-forms

(no tangential jump)

X is closed ie X corresponds tosimplicial 1-form

X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

this 1-form is co-closed(condition at vertices)X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

this 1-form is closed(condition at faces)X is pcw cnst

this 1-form is co-closed(condition at vertices)X is co-closed

this 1-form is co-closed(condition at vertices)

this 1-form is closed(condition at faces)X is pcw cnst harmonic 1-froms on graphs

cf GortlerGotsmanThurston

X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

(watch out cotan edge weightscould be negative cf BobenkoSpringborn forgetting non-negative weights by intrinsic edge flips)

Discrete Hodge Star

Complex multiplication (rotation by )

Discrete Hodge Star

X has no normal jump across edges

Complex multiplication (rotation by )

hellipdoes not take to itself

Because

X has no tangential jump across edges

X would have to be globally continuous

Discrete Hodge Star

Complex multiplication (rotation by )

Hodge star

hellipin other words Hodge star of X is harmonic part of J(X)

hellipdoes not take to itself

Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

(image are piecewise linear forms)

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

Adjoint operator

yields Whitney-Hodge decomposition

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57) by construction

deRham map DodziukPatodirsquo76

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of area

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Convergence of Hodge decomposition

Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)

bull This remains valid for variable (and converging) metrics

bull FEM-Whitney relations

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Thank you

• Algebraic Topology on Polyhedrafrom Linear Finite Elements
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Harmonic vector fields
• Contents
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Contents
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Contents
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Contents
• Discrete Harmonic vector fields
• Discrete Harmonic vector fields
• Computing harmonic fields
• Computing harmonic fields
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Discrete Hodge Star
• Discrete Hodge Star
• Discrete Hodge Star
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• Contents
• Convergence What can go wrong
• Convergence What can go wrong
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition

Discrete deRham

space of continuous PL functions on polyhedron

Discrete gradient

bull Gradients are piecewise constant

Discrete deRham

space of continuous PL functions on polyhedron

Discrete gradient

bull Gradients are piecewise constantbull tangential components are equal

along any edge

Discrete deRham

space of continuous PL functions on polyhedron

bull Gradients are piecewise constantbull tangential components are equal

along any edge

Local integrability iff locally

Discrete gradient

Discrete deRham

Discrete curl

(tangential jump)

ifftangential components are equalalong any edge

Discrete deRham

(tangential jump)

PL functionson vertices

PL functionson edge midpoints

PC vectorfields

Discrete deRham complex

ifftangential components are equalalong any edge

Discrete deRham

Lagrangeelement

Crouzeix-Raviartrsquo73

Discrete deRham

Lagrangeelement

Crouzeix-Raviartrsquo73

If then complex is exact in middle comp

bull ArnoldFalkrsquo89 ldquoA uniformly accurate FEM for Reissner-Mindlin Platerdquobull PolthierPreussrsquo03 ldquoIdentifying Vector Field Singularitieshelliprdquo

Discrete deRham

In fact complex yields correct cohomology

(dim = 2genus)

(not surprising - think simplicial = closed simplicial 1-forms)

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Discrete Hodge for closed meshes

(adjoint complex with respect to Lsup2 inner products)

Discrete Hodge for closed meshes

Hodge decomposition is simple linear algebra (since dim is finite)(B Eckmannrsquo44 ldquoHarmonische Funktionenhelliprdquo)

Discrete Hodge for closed meshes

harmonic vector fieldsDiscrete Hodge decomposition

Discrete Hodge for closed meshes

harmonic vector fieldsDiscrete Hodge decomposition

Lemma Decomposition only depends on choice of spaces and inner product on It does neither

depend on inner products on and nor on choice of operators

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Discrete Harmonic vector fields

dim = 2genus

Discrete Harmonic vector fields

Solve sparse systemfor u and v

dim = 2genus

Get method for computing

Computing harmonic fields

dim = 2genus

1 Init H = 0 2 Pick random PC vector field X3 Compute potential and co-potential of X4 Compute harmonic part h of X5 h in H IF TRUE goto 2 ELSE H = span(Hh)6 dim(H) = 2genus IF TRUE exit ELSE goto 2

Complexity (4genus) sparse systems (because odds that step 5 returns TRUE are zero)

Las Vegas algorithm for computing harmonic fields

Computing harmonic fields

Harmonic fields cont

Harmonic fields cont

no tangential jump

X is closed

Harmonic fields cont

no tangential jump

X is co-closed

X is closed

Harmonic fields amp harmonic1-forms

(no tangential jump)

X is closed ie X corresponds tosimplicial 1-form

X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

this 1-form is co-closed(condition at vertices)X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

this 1-form is closed(condition at faces)X is pcw cnst

this 1-form is co-closed(condition at vertices)X is co-closed

this 1-form is co-closed(condition at vertices)

this 1-form is closed(condition at faces)X is pcw cnst harmonic 1-froms on graphs

cf GortlerGotsmanThurston

X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

(watch out cotan edge weightscould be negative cf BobenkoSpringborn forgetting non-negative weights by intrinsic edge flips)

Discrete Hodge Star

Complex multiplication (rotation by )

Discrete Hodge Star

X has no normal jump across edges

Complex multiplication (rotation by )

hellipdoes not take to itself

Because

X has no tangential jump across edges

X would have to be globally continuous

Discrete Hodge Star

Complex multiplication (rotation by )

Hodge star

hellipin other words Hodge star of X is harmonic part of J(X)

hellipdoes not take to itself

Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

(image are piecewise linear forms)

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

Adjoint operator

yields Whitney-Hodge decomposition

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57) by construction

deRham map DodziukPatodirsquo76

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of area

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Convergence of Hodge decomposition

Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)

bull This remains valid for variable (and converging) metrics

bull FEM-Whitney relations

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Thank you

• Algebraic Topology on Polyhedrafrom Linear Finite Elements
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Harmonic vector fields
• Contents
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Contents
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Contents
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Contents
• Discrete Harmonic vector fields
• Discrete Harmonic vector fields
• Computing harmonic fields
• Computing harmonic fields
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Discrete Hodge Star
• Discrete Hodge Star
• Discrete Hodge Star
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• Contents
• Convergence What can go wrong
• Convergence What can go wrong
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition

Discrete deRham

space of continuous PL functions on polyhedron

Discrete gradient

bull Gradients are piecewise constantbull tangential components are equal

along any edge

Discrete deRham

space of continuous PL functions on polyhedron

bull Gradients are piecewise constantbull tangential components are equal

along any edge

Local integrability iff locally

Discrete gradient

Discrete deRham

Discrete curl

(tangential jump)

ifftangential components are equalalong any edge

Discrete deRham

(tangential jump)

PL functionson vertices

PL functionson edge midpoints

PC vectorfields

Discrete deRham complex

ifftangential components are equalalong any edge

Discrete deRham

Lagrangeelement

Crouzeix-Raviartrsquo73

Discrete deRham

Lagrangeelement

Crouzeix-Raviartrsquo73

If then complex is exact in middle comp

bull ArnoldFalkrsquo89 ldquoA uniformly accurate FEM for Reissner-Mindlin Platerdquobull PolthierPreussrsquo03 ldquoIdentifying Vector Field Singularitieshelliprdquo

Discrete deRham

In fact complex yields correct cohomology

(dim = 2genus)

(not surprising - think simplicial = closed simplicial 1-forms)

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Discrete Hodge for closed meshes

(adjoint complex with respect to Lsup2 inner products)

Discrete Hodge for closed meshes

Hodge decomposition is simple linear algebra (since dim is finite)(B Eckmannrsquo44 ldquoHarmonische Funktionenhelliprdquo)

Discrete Hodge for closed meshes

harmonic vector fieldsDiscrete Hodge decomposition

Discrete Hodge for closed meshes

harmonic vector fieldsDiscrete Hodge decomposition

Lemma Decomposition only depends on choice of spaces and inner product on It does neither

depend on inner products on and nor on choice of operators

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Discrete Harmonic vector fields

dim = 2genus

Discrete Harmonic vector fields

Solve sparse systemfor u and v

dim = 2genus

Get method for computing

Computing harmonic fields

dim = 2genus

1 Init H = 0 2 Pick random PC vector field X3 Compute potential and co-potential of X4 Compute harmonic part h of X5 h in H IF TRUE goto 2 ELSE H = span(Hh)6 dim(H) = 2genus IF TRUE exit ELSE goto 2

Complexity (4genus) sparse systems (because odds that step 5 returns TRUE are zero)

Las Vegas algorithm for computing harmonic fields

Computing harmonic fields

Harmonic fields cont

Harmonic fields cont

no tangential jump

X is closed

Harmonic fields cont

no tangential jump

X is co-closed

X is closed

Harmonic fields amp harmonic1-forms

(no tangential jump)

X is closed ie X corresponds tosimplicial 1-form

X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

this 1-form is co-closed(condition at vertices)X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

this 1-form is closed(condition at faces)X is pcw cnst

this 1-form is co-closed(condition at vertices)X is co-closed

this 1-form is co-closed(condition at vertices)

this 1-form is closed(condition at faces)X is pcw cnst harmonic 1-froms on graphs

cf GortlerGotsmanThurston

X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

(watch out cotan edge weightscould be negative cf BobenkoSpringborn forgetting non-negative weights by intrinsic edge flips)

Discrete Hodge Star

Complex multiplication (rotation by )

Discrete Hodge Star

X has no normal jump across edges

Complex multiplication (rotation by )

hellipdoes not take to itself

Because

X has no tangential jump across edges

X would have to be globally continuous

Discrete Hodge Star

Complex multiplication (rotation by )

Hodge star

hellipin other words Hodge star of X is harmonic part of J(X)

hellipdoes not take to itself

Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

(image are piecewise linear forms)

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

Adjoint operator

yields Whitney-Hodge decomposition

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57) by construction

deRham map DodziukPatodirsquo76

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of area

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Convergence of Hodge decomposition

Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)

bull This remains valid for variable (and converging) metrics

bull FEM-Whitney relations

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Thank you

• Algebraic Topology on Polyhedrafrom Linear Finite Elements
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Harmonic vector fields
• Contents
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Contents
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Contents
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Contents
• Discrete Harmonic vector fields
• Discrete Harmonic vector fields
• Computing harmonic fields
• Computing harmonic fields
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Discrete Hodge Star
• Discrete Hodge Star
• Discrete Hodge Star
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• Contents
• Convergence What can go wrong
• Convergence What can go wrong
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition

Discrete deRham

space of continuous PL functions on polyhedron

bull Gradients are piecewise constantbull tangential components are equal

along any edge

Local integrability iff locally

Discrete gradient

Discrete deRham

Discrete curl

(tangential jump)

ifftangential components are equalalong any edge

Discrete deRham

(tangential jump)

PL functionson vertices

PL functionson edge midpoints

PC vectorfields

Discrete deRham complex

ifftangential components are equalalong any edge

Discrete deRham

Lagrangeelement

Crouzeix-Raviartrsquo73

Discrete deRham

Lagrangeelement

Crouzeix-Raviartrsquo73

If then complex is exact in middle comp

bull ArnoldFalkrsquo89 ldquoA uniformly accurate FEM for Reissner-Mindlin Platerdquobull PolthierPreussrsquo03 ldquoIdentifying Vector Field Singularitieshelliprdquo

Discrete deRham

In fact complex yields correct cohomology

(dim = 2genus)

(not surprising - think simplicial = closed simplicial 1-forms)

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Discrete Hodge for closed meshes

(adjoint complex with respect to Lsup2 inner products)

Discrete Hodge for closed meshes

Hodge decomposition is simple linear algebra (since dim is finite)(B Eckmannrsquo44 ldquoHarmonische Funktionenhelliprdquo)

Discrete Hodge for closed meshes

harmonic vector fieldsDiscrete Hodge decomposition

Discrete Hodge for closed meshes

harmonic vector fieldsDiscrete Hodge decomposition

Lemma Decomposition only depends on choice of spaces and inner product on It does neither

depend on inner products on and nor on choice of operators

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Discrete Harmonic vector fields

dim = 2genus

Discrete Harmonic vector fields

Solve sparse systemfor u and v

dim = 2genus

Get method for computing

Computing harmonic fields

dim = 2genus

1 Init H = 0 2 Pick random PC vector field X3 Compute potential and co-potential of X4 Compute harmonic part h of X5 h in H IF TRUE goto 2 ELSE H = span(Hh)6 dim(H) = 2genus IF TRUE exit ELSE goto 2

Complexity (4genus) sparse systems (because odds that step 5 returns TRUE are zero)

Las Vegas algorithm for computing harmonic fields

Computing harmonic fields

Harmonic fields cont

Harmonic fields cont

no tangential jump

X is closed

Harmonic fields cont

no tangential jump

X is co-closed

X is closed

Harmonic fields amp harmonic1-forms

(no tangential jump)

X is closed ie X corresponds tosimplicial 1-form

X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

this 1-form is co-closed(condition at vertices)X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

this 1-form is closed(condition at faces)X is pcw cnst

this 1-form is co-closed(condition at vertices)X is co-closed

this 1-form is co-closed(condition at vertices)

this 1-form is closed(condition at faces)X is pcw cnst harmonic 1-froms on graphs

cf GortlerGotsmanThurston

X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

(watch out cotan edge weightscould be negative cf BobenkoSpringborn forgetting non-negative weights by intrinsic edge flips)

Discrete Hodge Star

Complex multiplication (rotation by )

Discrete Hodge Star

X has no normal jump across edges

Complex multiplication (rotation by )

hellipdoes not take to itself

Because

X has no tangential jump across edges

X would have to be globally continuous

Discrete Hodge Star

Complex multiplication (rotation by )

Hodge star

hellipin other words Hodge star of X is harmonic part of J(X)

hellipdoes not take to itself

Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

(image are piecewise linear forms)

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

Adjoint operator

yields Whitney-Hodge decomposition

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57) by construction

deRham map DodziukPatodirsquo76

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of area

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Convergence of Hodge decomposition

Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)

bull This remains valid for variable (and converging) metrics

bull FEM-Whitney relations

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Thank you

• Algebraic Topology on Polyhedrafrom Linear Finite Elements
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Harmonic vector fields
• Contents
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Contents
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Contents
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Contents
• Discrete Harmonic vector fields
• Discrete Harmonic vector fields
• Computing harmonic fields
• Computing harmonic fields
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Discrete Hodge Star
• Discrete Hodge Star
• Discrete Hodge Star
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• Contents
• Convergence What can go wrong
• Convergence What can go wrong
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition

Discrete deRham

Discrete curl

(tangential jump)

ifftangential components are equalalong any edge

Discrete deRham

(tangential jump)

PL functionson vertices

PL functionson edge midpoints

PC vectorfields

Discrete deRham complex

ifftangential components are equalalong any edge

Discrete deRham

Lagrangeelement

Crouzeix-Raviartrsquo73

Discrete deRham

Lagrangeelement

Crouzeix-Raviartrsquo73

If then complex is exact in middle comp

bull ArnoldFalkrsquo89 ldquoA uniformly accurate FEM for Reissner-Mindlin Platerdquobull PolthierPreussrsquo03 ldquoIdentifying Vector Field Singularitieshelliprdquo

Discrete deRham

In fact complex yields correct cohomology

(dim = 2genus)

(not surprising - think simplicial = closed simplicial 1-forms)

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Discrete Hodge for closed meshes

(adjoint complex with respect to Lsup2 inner products)

Discrete Hodge for closed meshes

Hodge decomposition is simple linear algebra (since dim is finite)(B Eckmannrsquo44 ldquoHarmonische Funktionenhelliprdquo)

Discrete Hodge for closed meshes

harmonic vector fieldsDiscrete Hodge decomposition

Discrete Hodge for closed meshes

harmonic vector fieldsDiscrete Hodge decomposition

Lemma Decomposition only depends on choice of spaces and inner product on It does neither

depend on inner products on and nor on choice of operators

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Discrete Harmonic vector fields

dim = 2genus

Discrete Harmonic vector fields

Solve sparse systemfor u and v

dim = 2genus

Get method for computing

Computing harmonic fields

dim = 2genus

1 Init H = 0 2 Pick random PC vector field X3 Compute potential and co-potential of X4 Compute harmonic part h of X5 h in H IF TRUE goto 2 ELSE H = span(Hh)6 dim(H) = 2genus IF TRUE exit ELSE goto 2

Complexity (4genus) sparse systems (because odds that step 5 returns TRUE are zero)

Las Vegas algorithm for computing harmonic fields

Computing harmonic fields

Harmonic fields cont

Harmonic fields cont

no tangential jump

X is closed

Harmonic fields cont

no tangential jump

X is co-closed

X is closed

Harmonic fields amp harmonic1-forms

(no tangential jump)

X is closed ie X corresponds tosimplicial 1-form

X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

this 1-form is co-closed(condition at vertices)X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

this 1-form is closed(condition at faces)X is pcw cnst

this 1-form is co-closed(condition at vertices)X is co-closed

this 1-form is co-closed(condition at vertices)

this 1-form is closed(condition at faces)X is pcw cnst harmonic 1-froms on graphs

cf GortlerGotsmanThurston

X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

(watch out cotan edge weightscould be negative cf BobenkoSpringborn forgetting non-negative weights by intrinsic edge flips)

Discrete Hodge Star

Complex multiplication (rotation by )

Discrete Hodge Star

X has no normal jump across edges

Complex multiplication (rotation by )

hellipdoes not take to itself

Because

X has no tangential jump across edges

X would have to be globally continuous

Discrete Hodge Star

Complex multiplication (rotation by )

Hodge star

hellipin other words Hodge star of X is harmonic part of J(X)

hellipdoes not take to itself

Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

(image are piecewise linear forms)

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

Adjoint operator

yields Whitney-Hodge decomposition

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57) by construction

deRham map DodziukPatodirsquo76

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of area

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Convergence of Hodge decomposition

Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)

bull This remains valid for variable (and converging) metrics

bull FEM-Whitney relations

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Thank you

• Algebraic Topology on Polyhedrafrom Linear Finite Elements
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Harmonic vector fields
• Contents
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Contents
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Contents
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Contents
• Discrete Harmonic vector fields
• Discrete Harmonic vector fields
• Computing harmonic fields
• Computing harmonic fields
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Discrete Hodge Star
• Discrete Hodge Star
• Discrete Hodge Star
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• Contents
• Convergence What can go wrong
• Convergence What can go wrong
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition

Discrete deRham

(tangential jump)

PL functionson vertices

PL functionson edge midpoints

PC vectorfields

Discrete deRham complex

ifftangential components are equalalong any edge

Discrete deRham

Lagrangeelement

Crouzeix-Raviartrsquo73

Discrete deRham

Lagrangeelement

Crouzeix-Raviartrsquo73

If then complex is exact in middle comp

bull ArnoldFalkrsquo89 ldquoA uniformly accurate FEM for Reissner-Mindlin Platerdquobull PolthierPreussrsquo03 ldquoIdentifying Vector Field Singularitieshelliprdquo

Discrete deRham

In fact complex yields correct cohomology

(dim = 2genus)

(not surprising - think simplicial = closed simplicial 1-forms)

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Discrete Hodge for closed meshes

(adjoint complex with respect to Lsup2 inner products)

Discrete Hodge for closed meshes

Hodge decomposition is simple linear algebra (since dim is finite)(B Eckmannrsquo44 ldquoHarmonische Funktionenhelliprdquo)

Discrete Hodge for closed meshes

harmonic vector fieldsDiscrete Hodge decomposition

Discrete Hodge for closed meshes

harmonic vector fieldsDiscrete Hodge decomposition

Lemma Decomposition only depends on choice of spaces and inner product on It does neither

depend on inner products on and nor on choice of operators

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Discrete Harmonic vector fields

dim = 2genus

Discrete Harmonic vector fields

Solve sparse systemfor u and v

dim = 2genus

Get method for computing

Computing harmonic fields

dim = 2genus

1 Init H = 0 2 Pick random PC vector field X3 Compute potential and co-potential of X4 Compute harmonic part h of X5 h in H IF TRUE goto 2 ELSE H = span(Hh)6 dim(H) = 2genus IF TRUE exit ELSE goto 2

Complexity (4genus) sparse systems (because odds that step 5 returns TRUE are zero)

Las Vegas algorithm for computing harmonic fields

Computing harmonic fields

Harmonic fields cont

Harmonic fields cont

no tangential jump

X is closed

Harmonic fields cont

no tangential jump

X is co-closed

X is closed

Harmonic fields amp harmonic1-forms

(no tangential jump)

X is closed ie X corresponds tosimplicial 1-form

X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

this 1-form is co-closed(condition at vertices)X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

this 1-form is closed(condition at faces)X is pcw cnst

this 1-form is co-closed(condition at vertices)X is co-closed

this 1-form is co-closed(condition at vertices)

this 1-form is closed(condition at faces)X is pcw cnst harmonic 1-froms on graphs

cf GortlerGotsmanThurston

X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

(watch out cotan edge weightscould be negative cf BobenkoSpringborn forgetting non-negative weights by intrinsic edge flips)

Discrete Hodge Star

Complex multiplication (rotation by )

Discrete Hodge Star

X has no normal jump across edges

Complex multiplication (rotation by )

hellipdoes not take to itself

Because

X has no tangential jump across edges

X would have to be globally continuous

Discrete Hodge Star

Complex multiplication (rotation by )

Hodge star

hellipin other words Hodge star of X is harmonic part of J(X)

hellipdoes not take to itself

Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

(image are piecewise linear forms)

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

Adjoint operator

yields Whitney-Hodge decomposition

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57) by construction

deRham map DodziukPatodirsquo76

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of area

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Convergence of Hodge decomposition

Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)

bull This remains valid for variable (and converging) metrics

bull FEM-Whitney relations

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Thank you

• Algebraic Topology on Polyhedrafrom Linear Finite Elements
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Harmonic vector fields
• Contents
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Contents
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Contents
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Contents
• Discrete Harmonic vector fields
• Discrete Harmonic vector fields
• Computing harmonic fields
• Computing harmonic fields
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Discrete Hodge Star
• Discrete Hodge Star
• Discrete Hodge Star
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• Contents
• Convergence What can go wrong
• Convergence What can go wrong
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition

Discrete deRham

Lagrangeelement

Crouzeix-Raviartrsquo73

Discrete deRham

Lagrangeelement

Crouzeix-Raviartrsquo73

If then complex is exact in middle comp

bull ArnoldFalkrsquo89 ldquoA uniformly accurate FEM for Reissner-Mindlin Platerdquobull PolthierPreussrsquo03 ldquoIdentifying Vector Field Singularitieshelliprdquo

Discrete deRham

In fact complex yields correct cohomology

(dim = 2genus)

(not surprising - think simplicial = closed simplicial 1-forms)

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Discrete Hodge for closed meshes

(adjoint complex with respect to Lsup2 inner products)

Discrete Hodge for closed meshes

Hodge decomposition is simple linear algebra (since dim is finite)(B Eckmannrsquo44 ldquoHarmonische Funktionenhelliprdquo)

Discrete Hodge for closed meshes

harmonic vector fieldsDiscrete Hodge decomposition

Discrete Hodge for closed meshes

harmonic vector fieldsDiscrete Hodge decomposition

Lemma Decomposition only depends on choice of spaces and inner product on It does neither

depend on inner products on and nor on choice of operators

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Discrete Harmonic vector fields

dim = 2genus

Discrete Harmonic vector fields

Solve sparse systemfor u and v

dim = 2genus

Get method for computing

Computing harmonic fields

dim = 2genus

1 Init H = 0 2 Pick random PC vector field X3 Compute potential and co-potential of X4 Compute harmonic part h of X5 h in H IF TRUE goto 2 ELSE H = span(Hh)6 dim(H) = 2genus IF TRUE exit ELSE goto 2

Complexity (4genus) sparse systems (because odds that step 5 returns TRUE are zero)

Las Vegas algorithm for computing harmonic fields

Computing harmonic fields

Harmonic fields cont

Harmonic fields cont

no tangential jump

X is closed

Harmonic fields cont

no tangential jump

X is co-closed

X is closed

Harmonic fields amp harmonic1-forms

(no tangential jump)

X is closed ie X corresponds tosimplicial 1-form

X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

this 1-form is co-closed(condition at vertices)X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

this 1-form is closed(condition at faces)X is pcw cnst

this 1-form is co-closed(condition at vertices)X is co-closed

this 1-form is co-closed(condition at vertices)

this 1-form is closed(condition at faces)X is pcw cnst harmonic 1-froms on graphs

cf GortlerGotsmanThurston

X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

(watch out cotan edge weightscould be negative cf BobenkoSpringborn forgetting non-negative weights by intrinsic edge flips)

Discrete Hodge Star

Complex multiplication (rotation by )

Discrete Hodge Star

X has no normal jump across edges

Complex multiplication (rotation by )

hellipdoes not take to itself

Because

X has no tangential jump across edges

X would have to be globally continuous

Discrete Hodge Star

Complex multiplication (rotation by )

Hodge star

hellipin other words Hodge star of X is harmonic part of J(X)

hellipdoes not take to itself

Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

(image are piecewise linear forms)

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

Adjoint operator

yields Whitney-Hodge decomposition

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57) by construction

deRham map DodziukPatodirsquo76

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of area

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Convergence of Hodge decomposition

Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)

bull This remains valid for variable (and converging) metrics

bull FEM-Whitney relations

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Thank you

• Algebraic Topology on Polyhedrafrom Linear Finite Elements
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Harmonic vector fields
• Contents
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Contents
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Contents
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Contents
• Discrete Harmonic vector fields
• Discrete Harmonic vector fields
• Computing harmonic fields
• Computing harmonic fields
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Discrete Hodge Star
• Discrete Hodge Star
• Discrete Hodge Star
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• Contents
• Convergence What can go wrong
• Convergence What can go wrong
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition

Discrete deRham

Lagrangeelement

Crouzeix-Raviartrsquo73

If then complex is exact in middle comp

bull ArnoldFalkrsquo89 ldquoA uniformly accurate FEM for Reissner-Mindlin Platerdquobull PolthierPreussrsquo03 ldquoIdentifying Vector Field Singularitieshelliprdquo

Discrete deRham

In fact complex yields correct cohomology

(dim = 2genus)

(not surprising - think simplicial = closed simplicial 1-forms)

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Discrete Hodge for closed meshes

(adjoint complex with respect to Lsup2 inner products)

Discrete Hodge for closed meshes

Hodge decomposition is simple linear algebra (since dim is finite)(B Eckmannrsquo44 ldquoHarmonische Funktionenhelliprdquo)

Discrete Hodge for closed meshes

harmonic vector fieldsDiscrete Hodge decomposition

Discrete Hodge for closed meshes

harmonic vector fieldsDiscrete Hodge decomposition

Lemma Decomposition only depends on choice of spaces and inner product on It does neither

depend on inner products on and nor on choice of operators

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Discrete Harmonic vector fields

dim = 2genus

Discrete Harmonic vector fields

Solve sparse systemfor u and v

dim = 2genus

Get method for computing

Computing harmonic fields

dim = 2genus

1 Init H = 0 2 Pick random PC vector field X3 Compute potential and co-potential of X4 Compute harmonic part h of X5 h in H IF TRUE goto 2 ELSE H = span(Hh)6 dim(H) = 2genus IF TRUE exit ELSE goto 2

Complexity (4genus) sparse systems (because odds that step 5 returns TRUE are zero)

Las Vegas algorithm for computing harmonic fields

Computing harmonic fields

Harmonic fields cont

Harmonic fields cont

no tangential jump

X is closed

Harmonic fields cont

no tangential jump

X is co-closed

X is closed

Harmonic fields amp harmonic1-forms

(no tangential jump)

X is closed ie X corresponds tosimplicial 1-form

X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

this 1-form is co-closed(condition at vertices)X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

this 1-form is closed(condition at faces)X is pcw cnst

this 1-form is co-closed(condition at vertices)X is co-closed

this 1-form is co-closed(condition at vertices)

this 1-form is closed(condition at faces)X is pcw cnst harmonic 1-froms on graphs

cf GortlerGotsmanThurston

X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

(watch out cotan edge weightscould be negative cf BobenkoSpringborn forgetting non-negative weights by intrinsic edge flips)

Discrete Hodge Star

Complex multiplication (rotation by )

Discrete Hodge Star

X has no normal jump across edges

Complex multiplication (rotation by )

hellipdoes not take to itself

Because

X has no tangential jump across edges

X would have to be globally continuous

Discrete Hodge Star

Complex multiplication (rotation by )

Hodge star

hellipin other words Hodge star of X is harmonic part of J(X)

hellipdoes not take to itself

Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

(image are piecewise linear forms)

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

Adjoint operator

yields Whitney-Hodge decomposition

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57) by construction

deRham map DodziukPatodirsquo76

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of area

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Convergence of Hodge decomposition

Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)

bull This remains valid for variable (and converging) metrics

bull FEM-Whitney relations

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Thank you

• Algebraic Topology on Polyhedrafrom Linear Finite Elements
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Harmonic vector fields
• Contents
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Contents
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Contents
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Contents
• Discrete Harmonic vector fields
• Discrete Harmonic vector fields
• Computing harmonic fields
• Computing harmonic fields
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Discrete Hodge Star
• Discrete Hodge Star
• Discrete Hodge Star
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• Contents
• Convergence What can go wrong
• Convergence What can go wrong
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition

Discrete deRham

In fact complex yields correct cohomology

(dim = 2genus)

(not surprising - think simplicial = closed simplicial 1-forms)

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Discrete Hodge for closed meshes

(adjoint complex with respect to Lsup2 inner products)

Discrete Hodge for closed meshes

Hodge decomposition is simple linear algebra (since dim is finite)(B Eckmannrsquo44 ldquoHarmonische Funktionenhelliprdquo)

Discrete Hodge for closed meshes

harmonic vector fieldsDiscrete Hodge decomposition

Discrete Hodge for closed meshes

harmonic vector fieldsDiscrete Hodge decomposition

Lemma Decomposition only depends on choice of spaces and inner product on It does neither

depend on inner products on and nor on choice of operators

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Discrete Harmonic vector fields

dim = 2genus

Discrete Harmonic vector fields

Solve sparse systemfor u and v

dim = 2genus

Get method for computing

Computing harmonic fields

dim = 2genus

1 Init H = 0 2 Pick random PC vector field X3 Compute potential and co-potential of X4 Compute harmonic part h of X5 h in H IF TRUE goto 2 ELSE H = span(Hh)6 dim(H) = 2genus IF TRUE exit ELSE goto 2

Complexity (4genus) sparse systems (because odds that step 5 returns TRUE are zero)

Las Vegas algorithm for computing harmonic fields

Computing harmonic fields

Harmonic fields cont

Harmonic fields cont

no tangential jump

X is closed

Harmonic fields cont

no tangential jump

X is co-closed

X is closed

Harmonic fields amp harmonic1-forms

(no tangential jump)

X is closed ie X corresponds tosimplicial 1-form

X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

this 1-form is co-closed(condition at vertices)X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

this 1-form is closed(condition at faces)X is pcw cnst

this 1-form is co-closed(condition at vertices)X is co-closed

this 1-form is co-closed(condition at vertices)

this 1-form is closed(condition at faces)X is pcw cnst harmonic 1-froms on graphs

cf GortlerGotsmanThurston

X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

(watch out cotan edge weightscould be negative cf BobenkoSpringborn forgetting non-negative weights by intrinsic edge flips)

Discrete Hodge Star

Complex multiplication (rotation by )

Discrete Hodge Star

X has no normal jump across edges

Complex multiplication (rotation by )

hellipdoes not take to itself

Because

X has no tangential jump across edges

X would have to be globally continuous

Discrete Hodge Star

Complex multiplication (rotation by )

Hodge star

hellipin other words Hodge star of X is harmonic part of J(X)

hellipdoes not take to itself

Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

(image are piecewise linear forms)

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

Adjoint operator

yields Whitney-Hodge decomposition

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57) by construction

deRham map DodziukPatodirsquo76

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of area

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Convergence of Hodge decomposition

Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)

bull This remains valid for variable (and converging) metrics

bull FEM-Whitney relations

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Thank you

• Algebraic Topology on Polyhedrafrom Linear Finite Elements
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Harmonic vector fields
• Contents
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Contents
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Contents
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Contents
• Discrete Harmonic vector fields
• Discrete Harmonic vector fields
• Computing harmonic fields
• Computing harmonic fields
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Discrete Hodge Star
• Discrete Hodge Star
• Discrete Hodge Star
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• Contents
• Convergence What can go wrong
• Convergence What can go wrong
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Discrete Hodge for closed meshes

(adjoint complex with respect to Lsup2 inner products)

Discrete Hodge for closed meshes

Hodge decomposition is simple linear algebra (since dim is finite)(B Eckmannrsquo44 ldquoHarmonische Funktionenhelliprdquo)

Discrete Hodge for closed meshes

harmonic vector fieldsDiscrete Hodge decomposition

Discrete Hodge for closed meshes

harmonic vector fieldsDiscrete Hodge decomposition

Lemma Decomposition only depends on choice of spaces and inner product on It does neither

depend on inner products on and nor on choice of operators

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Discrete Harmonic vector fields

dim = 2genus

Discrete Harmonic vector fields

Solve sparse systemfor u and v

dim = 2genus

Get method for computing

Computing harmonic fields

dim = 2genus

1 Init H = 0 2 Pick random PC vector field X3 Compute potential and co-potential of X4 Compute harmonic part h of X5 h in H IF TRUE goto 2 ELSE H = span(Hh)6 dim(H) = 2genus IF TRUE exit ELSE goto 2

Complexity (4genus) sparse systems (because odds that step 5 returns TRUE are zero)

Las Vegas algorithm for computing harmonic fields

Computing harmonic fields

Harmonic fields cont

Harmonic fields cont

no tangential jump

X is closed

Harmonic fields cont

no tangential jump

X is co-closed

X is closed

Harmonic fields amp harmonic1-forms

(no tangential jump)

X is closed ie X corresponds tosimplicial 1-form

X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

this 1-form is co-closed(condition at vertices)X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

this 1-form is closed(condition at faces)X is pcw cnst

this 1-form is co-closed(condition at vertices)X is co-closed

this 1-form is co-closed(condition at vertices)

this 1-form is closed(condition at faces)X is pcw cnst harmonic 1-froms on graphs

cf GortlerGotsmanThurston

X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

(watch out cotan edge weightscould be negative cf BobenkoSpringborn forgetting non-negative weights by intrinsic edge flips)

Discrete Hodge Star

Complex multiplication (rotation by )

Discrete Hodge Star

X has no normal jump across edges

Complex multiplication (rotation by )

hellipdoes not take to itself

Because

X has no tangential jump across edges

X would have to be globally continuous

Discrete Hodge Star

Complex multiplication (rotation by )

Hodge star

hellipin other words Hodge star of X is harmonic part of J(X)

hellipdoes not take to itself

Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

(image are piecewise linear forms)

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

Adjoint operator

yields Whitney-Hodge decomposition

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57) by construction

deRham map DodziukPatodirsquo76

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of area

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Convergence of Hodge decomposition

Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)

bull This remains valid for variable (and converging) metrics

bull FEM-Whitney relations

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Thank you

• Algebraic Topology on Polyhedrafrom Linear Finite Elements
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Harmonic vector fields
• Contents
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Contents
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Contents
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Contents
• Discrete Harmonic vector fields
• Discrete Harmonic vector fields
• Computing harmonic fields
• Computing harmonic fields
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Discrete Hodge Star
• Discrete Hodge Star
• Discrete Hodge Star
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• Contents
• Convergence What can go wrong
• Convergence What can go wrong
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition

Discrete Hodge for closed meshes

(adjoint complex with respect to Lsup2 inner products)

Discrete Hodge for closed meshes

Hodge decomposition is simple linear algebra (since dim is finite)(B Eckmannrsquo44 ldquoHarmonische Funktionenhelliprdquo)

Discrete Hodge for closed meshes

harmonic vector fieldsDiscrete Hodge decomposition

Discrete Hodge for closed meshes

harmonic vector fieldsDiscrete Hodge decomposition

Lemma Decomposition only depends on choice of spaces and inner product on It does neither

depend on inner products on and nor on choice of operators

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Discrete Harmonic vector fields

dim = 2genus

Discrete Harmonic vector fields

Solve sparse systemfor u and v

dim = 2genus

Get method for computing

Computing harmonic fields

dim = 2genus

1 Init H = 0 2 Pick random PC vector field X3 Compute potential and co-potential of X4 Compute harmonic part h of X5 h in H IF TRUE goto 2 ELSE H = span(Hh)6 dim(H) = 2genus IF TRUE exit ELSE goto 2

Complexity (4genus) sparse systems (because odds that step 5 returns TRUE are zero)

Las Vegas algorithm for computing harmonic fields

Computing harmonic fields

Harmonic fields cont

Harmonic fields cont

no tangential jump

X is closed

Harmonic fields cont

no tangential jump

X is co-closed

X is closed

Harmonic fields amp harmonic1-forms

(no tangential jump)

X is closed ie X corresponds tosimplicial 1-form

X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

this 1-form is co-closed(condition at vertices)X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

this 1-form is closed(condition at faces)X is pcw cnst

this 1-form is co-closed(condition at vertices)X is co-closed

this 1-form is co-closed(condition at vertices)

this 1-form is closed(condition at faces)X is pcw cnst harmonic 1-froms on graphs

cf GortlerGotsmanThurston

X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

(watch out cotan edge weightscould be negative cf BobenkoSpringborn forgetting non-negative weights by intrinsic edge flips)

Discrete Hodge Star

Complex multiplication (rotation by )

Discrete Hodge Star

X has no normal jump across edges

Complex multiplication (rotation by )

hellipdoes not take to itself

Because

X has no tangential jump across edges

X would have to be globally continuous

Discrete Hodge Star

Complex multiplication (rotation by )

Hodge star

hellipin other words Hodge star of X is harmonic part of J(X)

hellipdoes not take to itself

Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

(image are piecewise linear forms)

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

Adjoint operator

yields Whitney-Hodge decomposition

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57) by construction

deRham map DodziukPatodirsquo76

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of area

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Convergence of Hodge decomposition

Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)

bull This remains valid for variable (and converging) metrics

bull FEM-Whitney relations

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Thank you

• Algebraic Topology on Polyhedrafrom Linear Finite Elements
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Harmonic vector fields
• Contents
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Contents
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Contents
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Contents
• Discrete Harmonic vector fields
• Discrete Harmonic vector fields
• Computing harmonic fields
• Computing harmonic fields
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Discrete Hodge Star
• Discrete Hodge Star
• Discrete Hodge Star
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• Contents
• Convergence What can go wrong
• Convergence What can go wrong
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition

Discrete Hodge for closed meshes

Hodge decomposition is simple linear algebra (since dim is finite)(B Eckmannrsquo44 ldquoHarmonische Funktionenhelliprdquo)

Discrete Hodge for closed meshes

harmonic vector fieldsDiscrete Hodge decomposition

Discrete Hodge for closed meshes

harmonic vector fieldsDiscrete Hodge decomposition

Lemma Decomposition only depends on choice of spaces and inner product on It does neither

depend on inner products on and nor on choice of operators

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Discrete Harmonic vector fields

dim = 2genus

Discrete Harmonic vector fields

Solve sparse systemfor u and v

dim = 2genus

Get method for computing

Computing harmonic fields

dim = 2genus

1 Init H = 0 2 Pick random PC vector field X3 Compute potential and co-potential of X4 Compute harmonic part h of X5 h in H IF TRUE goto 2 ELSE H = span(Hh)6 dim(H) = 2genus IF TRUE exit ELSE goto 2

Complexity (4genus) sparse systems (because odds that step 5 returns TRUE are zero)

Las Vegas algorithm for computing harmonic fields

Computing harmonic fields

Harmonic fields cont

Harmonic fields cont

no tangential jump

X is closed

Harmonic fields cont

no tangential jump

X is co-closed

X is closed

Harmonic fields amp harmonic1-forms

(no tangential jump)

X is closed ie X corresponds tosimplicial 1-form

X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

this 1-form is co-closed(condition at vertices)X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

this 1-form is closed(condition at faces)X is pcw cnst

this 1-form is co-closed(condition at vertices)X is co-closed

this 1-form is co-closed(condition at vertices)

this 1-form is closed(condition at faces)X is pcw cnst harmonic 1-froms on graphs

cf GortlerGotsmanThurston

X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

(watch out cotan edge weightscould be negative cf BobenkoSpringborn forgetting non-negative weights by intrinsic edge flips)

Discrete Hodge Star

Complex multiplication (rotation by )

Discrete Hodge Star

X has no normal jump across edges

Complex multiplication (rotation by )

hellipdoes not take to itself

Because

X has no tangential jump across edges

X would have to be globally continuous

Discrete Hodge Star

Complex multiplication (rotation by )

Hodge star

hellipin other words Hodge star of X is harmonic part of J(X)

hellipdoes not take to itself

Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

(image are piecewise linear forms)

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

Adjoint operator

yields Whitney-Hodge decomposition

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57) by construction

deRham map DodziukPatodirsquo76

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of area

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Convergence of Hodge decomposition

Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)

bull This remains valid for variable (and converging) metrics

bull FEM-Whitney relations

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Thank you

• Algebraic Topology on Polyhedrafrom Linear Finite Elements
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Harmonic vector fields
• Contents
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Contents
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Contents
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Contents
• Discrete Harmonic vector fields
• Discrete Harmonic vector fields
• Computing harmonic fields
• Computing harmonic fields
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Discrete Hodge Star
• Discrete Hodge Star
• Discrete Hodge Star
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• Contents
• Convergence What can go wrong
• Convergence What can go wrong
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition

Discrete Hodge for closed meshes

harmonic vector fieldsDiscrete Hodge decomposition

Discrete Hodge for closed meshes

harmonic vector fieldsDiscrete Hodge decomposition

Lemma Decomposition only depends on choice of spaces and inner product on It does neither

depend on inner products on and nor on choice of operators

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Discrete Harmonic vector fields

dim = 2genus

Discrete Harmonic vector fields

Solve sparse systemfor u and v

dim = 2genus

Get method for computing

Computing harmonic fields

dim = 2genus

1 Init H = 0 2 Pick random PC vector field X3 Compute potential and co-potential of X4 Compute harmonic part h of X5 h in H IF TRUE goto 2 ELSE H = span(Hh)6 dim(H) = 2genus IF TRUE exit ELSE goto 2

Complexity (4genus) sparse systems (because odds that step 5 returns TRUE are zero)

Las Vegas algorithm for computing harmonic fields

Computing harmonic fields

Harmonic fields cont

Harmonic fields cont

no tangential jump

X is closed

Harmonic fields cont

no tangential jump

X is co-closed

X is closed

Harmonic fields amp harmonic1-forms

(no tangential jump)

X is closed ie X corresponds tosimplicial 1-form

X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

this 1-form is co-closed(condition at vertices)X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

this 1-form is closed(condition at faces)X is pcw cnst

this 1-form is co-closed(condition at vertices)X is co-closed

this 1-form is co-closed(condition at vertices)

this 1-form is closed(condition at faces)X is pcw cnst harmonic 1-froms on graphs

cf GortlerGotsmanThurston

X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

(watch out cotan edge weightscould be negative cf BobenkoSpringborn forgetting non-negative weights by intrinsic edge flips)

Discrete Hodge Star

Complex multiplication (rotation by )

Discrete Hodge Star

X has no normal jump across edges

Complex multiplication (rotation by )

hellipdoes not take to itself

Because

X has no tangential jump across edges

X would have to be globally continuous

Discrete Hodge Star

Complex multiplication (rotation by )

Hodge star

hellipin other words Hodge star of X is harmonic part of J(X)

hellipdoes not take to itself

Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

(image are piecewise linear forms)

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

Adjoint operator

yields Whitney-Hodge decomposition

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57) by construction

deRham map DodziukPatodirsquo76

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of area

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Convergence of Hodge decomposition

Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)

bull This remains valid for variable (and converging) metrics

bull FEM-Whitney relations

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Thank you

• Algebraic Topology on Polyhedrafrom Linear Finite Elements
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Harmonic vector fields
• Contents
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Contents
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Contents
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Contents
• Discrete Harmonic vector fields
• Discrete Harmonic vector fields
• Computing harmonic fields
• Computing harmonic fields
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Discrete Hodge Star
• Discrete Hodge Star
• Discrete Hodge Star
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• Contents
• Convergence What can go wrong
• Convergence What can go wrong
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition

Discrete Hodge for closed meshes

harmonic vector fieldsDiscrete Hodge decomposition

Lemma Decomposition only depends on choice of spaces and inner product on It does neither

depend on inner products on and nor on choice of operators

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Discrete Harmonic vector fields

dim = 2genus

Discrete Harmonic vector fields

Solve sparse systemfor u and v

dim = 2genus

Get method for computing

Computing harmonic fields

dim = 2genus

1 Init H = 0 2 Pick random PC vector field X3 Compute potential and co-potential of X4 Compute harmonic part h of X5 h in H IF TRUE goto 2 ELSE H = span(Hh)6 dim(H) = 2genus IF TRUE exit ELSE goto 2

Complexity (4genus) sparse systems (because odds that step 5 returns TRUE are zero)

Las Vegas algorithm for computing harmonic fields

Computing harmonic fields

Harmonic fields cont

Harmonic fields cont

no tangential jump

X is closed

Harmonic fields cont

no tangential jump

X is co-closed

X is closed

Harmonic fields amp harmonic1-forms

(no tangential jump)

X is closed ie X corresponds tosimplicial 1-form

X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

this 1-form is co-closed(condition at vertices)X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

this 1-form is closed(condition at faces)X is pcw cnst

this 1-form is co-closed(condition at vertices)X is co-closed

this 1-form is co-closed(condition at vertices)

this 1-form is closed(condition at faces)X is pcw cnst harmonic 1-froms on graphs

cf GortlerGotsmanThurston

X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

(watch out cotan edge weightscould be negative cf BobenkoSpringborn forgetting non-negative weights by intrinsic edge flips)

Discrete Hodge Star

Complex multiplication (rotation by )

Discrete Hodge Star

X has no normal jump across edges

Complex multiplication (rotation by )

hellipdoes not take to itself

Because

X has no tangential jump across edges

X would have to be globally continuous

Discrete Hodge Star

Complex multiplication (rotation by )

Hodge star

hellipin other words Hodge star of X is harmonic part of J(X)

hellipdoes not take to itself

Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

(image are piecewise linear forms)

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

Adjoint operator

yields Whitney-Hodge decomposition

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57) by construction

deRham map DodziukPatodirsquo76

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of area

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Convergence of Hodge decomposition

Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)

bull This remains valid for variable (and converging) metrics

bull FEM-Whitney relations

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Thank you

• Algebraic Topology on Polyhedrafrom Linear Finite Elements
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Harmonic vector fields
• Contents
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Contents
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Contents
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Contents
• Discrete Harmonic vector fields
• Discrete Harmonic vector fields
• Computing harmonic fields
• Computing harmonic fields
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Discrete Hodge Star
• Discrete Hodge Star
• Discrete Hodge Star
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• Contents
• Convergence What can go wrong
• Convergence What can go wrong
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Discrete Harmonic vector fields

dim = 2genus

Discrete Harmonic vector fields

Solve sparse systemfor u and v

dim = 2genus

Get method for computing

Computing harmonic fields

dim = 2genus

1 Init H = 0 2 Pick random PC vector field X3 Compute potential and co-potential of X4 Compute harmonic part h of X5 h in H IF TRUE goto 2 ELSE H = span(Hh)6 dim(H) = 2genus IF TRUE exit ELSE goto 2

Complexity (4genus) sparse systems (because odds that step 5 returns TRUE are zero)

Las Vegas algorithm for computing harmonic fields

Computing harmonic fields

Harmonic fields cont

Harmonic fields cont

no tangential jump

X is closed

Harmonic fields cont

no tangential jump

X is co-closed

X is closed

Harmonic fields amp harmonic1-forms

(no tangential jump)

X is closed ie X corresponds tosimplicial 1-form

X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

this 1-form is co-closed(condition at vertices)X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

this 1-form is closed(condition at faces)X is pcw cnst

this 1-form is co-closed(condition at vertices)X is co-closed

this 1-form is co-closed(condition at vertices)

this 1-form is closed(condition at faces)X is pcw cnst harmonic 1-froms on graphs

cf GortlerGotsmanThurston

X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

(watch out cotan edge weightscould be negative cf BobenkoSpringborn forgetting non-negative weights by intrinsic edge flips)

Discrete Hodge Star

Complex multiplication (rotation by )

Discrete Hodge Star

X has no normal jump across edges

Complex multiplication (rotation by )

hellipdoes not take to itself

Because

X has no tangential jump across edges

X would have to be globally continuous

Discrete Hodge Star

Complex multiplication (rotation by )

Hodge star

hellipin other words Hodge star of X is harmonic part of J(X)

hellipdoes not take to itself

Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

(image are piecewise linear forms)

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

Adjoint operator

yields Whitney-Hodge decomposition

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57) by construction

deRham map DodziukPatodirsquo76

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of area

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Convergence of Hodge decomposition

Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)

bull This remains valid for variable (and converging) metrics

bull FEM-Whitney relations

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Thank you

• Algebraic Topology on Polyhedrafrom Linear Finite Elements
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Harmonic vector fields
• Contents
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Contents
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Contents
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Contents
• Discrete Harmonic vector fields
• Discrete Harmonic vector fields
• Computing harmonic fields
• Computing harmonic fields
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Discrete Hodge Star
• Discrete Hodge Star
• Discrete Hodge Star
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• Contents
• Convergence What can go wrong
• Convergence What can go wrong
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition

Discrete Harmonic vector fields

dim = 2genus

Discrete Harmonic vector fields

Solve sparse systemfor u and v

dim = 2genus

Get method for computing

Computing harmonic fields

dim = 2genus

1 Init H = 0 2 Pick random PC vector field X3 Compute potential and co-potential of X4 Compute harmonic part h of X5 h in H IF TRUE goto 2 ELSE H = span(Hh)6 dim(H) = 2genus IF TRUE exit ELSE goto 2

Complexity (4genus) sparse systems (because odds that step 5 returns TRUE are zero)

Las Vegas algorithm for computing harmonic fields

Computing harmonic fields

Harmonic fields cont

Harmonic fields cont

no tangential jump

X is closed

Harmonic fields cont

no tangential jump

X is co-closed

X is closed

Harmonic fields amp harmonic1-forms

(no tangential jump)

X is closed ie X corresponds tosimplicial 1-form

X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

this 1-form is co-closed(condition at vertices)X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

this 1-form is closed(condition at faces)X is pcw cnst

this 1-form is co-closed(condition at vertices)X is co-closed

this 1-form is co-closed(condition at vertices)

this 1-form is closed(condition at faces)X is pcw cnst harmonic 1-froms on graphs

cf GortlerGotsmanThurston

X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

(watch out cotan edge weightscould be negative cf BobenkoSpringborn forgetting non-negative weights by intrinsic edge flips)

Discrete Hodge Star

Complex multiplication (rotation by )

Discrete Hodge Star

X has no normal jump across edges

Complex multiplication (rotation by )

hellipdoes not take to itself

Because

X has no tangential jump across edges

X would have to be globally continuous

Discrete Hodge Star

Complex multiplication (rotation by )

Hodge star

hellipin other words Hodge star of X is harmonic part of J(X)

hellipdoes not take to itself

Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

(image are piecewise linear forms)

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

Adjoint operator

yields Whitney-Hodge decomposition

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57) by construction

deRham map DodziukPatodirsquo76

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of area

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Convergence of Hodge decomposition

Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)

bull This remains valid for variable (and converging) metrics

bull FEM-Whitney relations

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Thank you

• Algebraic Topology on Polyhedrafrom Linear Finite Elements
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Harmonic vector fields
• Contents
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Contents
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Contents
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Contents
• Discrete Harmonic vector fields
• Discrete Harmonic vector fields
• Computing harmonic fields
• Computing harmonic fields
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Discrete Hodge Star
• Discrete Hodge Star
• Discrete Hodge Star
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• Contents
• Convergence What can go wrong
• Convergence What can go wrong
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition

Discrete Harmonic vector fields

Solve sparse systemfor u and v

dim = 2genus

Get method for computing

Computing harmonic fields

dim = 2genus

1 Init H = 0 2 Pick random PC vector field X3 Compute potential and co-potential of X4 Compute harmonic part h of X5 h in H IF TRUE goto 2 ELSE H = span(Hh)6 dim(H) = 2genus IF TRUE exit ELSE goto 2

Complexity (4genus) sparse systems (because odds that step 5 returns TRUE are zero)

Las Vegas algorithm for computing harmonic fields

Computing harmonic fields

Harmonic fields cont

Harmonic fields cont

no tangential jump

X is closed

Harmonic fields cont

no tangential jump

X is co-closed

X is closed

Harmonic fields amp harmonic1-forms

(no tangential jump)

X is closed ie X corresponds tosimplicial 1-form

X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

this 1-form is co-closed(condition at vertices)X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

this 1-form is closed(condition at faces)X is pcw cnst

this 1-form is co-closed(condition at vertices)X is co-closed

this 1-form is co-closed(condition at vertices)

this 1-form is closed(condition at faces)X is pcw cnst harmonic 1-froms on graphs

cf GortlerGotsmanThurston

X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

(watch out cotan edge weightscould be negative cf BobenkoSpringborn forgetting non-negative weights by intrinsic edge flips)

Discrete Hodge Star

Complex multiplication (rotation by )

Discrete Hodge Star

X has no normal jump across edges

Complex multiplication (rotation by )

hellipdoes not take to itself

Because

X has no tangential jump across edges

X would have to be globally continuous

Discrete Hodge Star

Complex multiplication (rotation by )

Hodge star

hellipin other words Hodge star of X is harmonic part of J(X)

hellipdoes not take to itself

Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

(image are piecewise linear forms)

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

Adjoint operator

yields Whitney-Hodge decomposition

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57) by construction

deRham map DodziukPatodirsquo76

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of area

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Convergence of Hodge decomposition

Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)

bull This remains valid for variable (and converging) metrics

bull FEM-Whitney relations

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Thank you

• Algebraic Topology on Polyhedrafrom Linear Finite Elements
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Harmonic vector fields
• Contents
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Contents
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Contents
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Contents
• Discrete Harmonic vector fields
• Discrete Harmonic vector fields
• Computing harmonic fields
• Computing harmonic fields
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Discrete Hodge Star
• Discrete Hodge Star
• Discrete Hodge Star
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• Contents
• Convergence What can go wrong
• Convergence What can go wrong
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition

Computing harmonic fields

dim = 2genus

1 Init H = 0 2 Pick random PC vector field X3 Compute potential and co-potential of X4 Compute harmonic part h of X5 h in H IF TRUE goto 2 ELSE H = span(Hh)6 dim(H) = 2genus IF TRUE exit ELSE goto 2

Complexity (4genus) sparse systems (because odds that step 5 returns TRUE are zero)

Las Vegas algorithm for computing harmonic fields

Computing harmonic fields

Harmonic fields cont

Harmonic fields cont

no tangential jump

X is closed

Harmonic fields cont

no tangential jump

X is co-closed

X is closed

Harmonic fields amp harmonic1-forms

(no tangential jump)

X is closed ie X corresponds tosimplicial 1-form

X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

this 1-form is co-closed(condition at vertices)X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

this 1-form is closed(condition at faces)X is pcw cnst

this 1-form is co-closed(condition at vertices)X is co-closed

this 1-form is co-closed(condition at vertices)

this 1-form is closed(condition at faces)X is pcw cnst harmonic 1-froms on graphs

cf GortlerGotsmanThurston

X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

(watch out cotan edge weightscould be negative cf BobenkoSpringborn forgetting non-negative weights by intrinsic edge flips)

Discrete Hodge Star

Complex multiplication (rotation by )

Discrete Hodge Star

X has no normal jump across edges

Complex multiplication (rotation by )

hellipdoes not take to itself

Because

X has no tangential jump across edges

X would have to be globally continuous

Discrete Hodge Star

Complex multiplication (rotation by )

Hodge star

hellipin other words Hodge star of X is harmonic part of J(X)

hellipdoes not take to itself

Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

(image are piecewise linear forms)

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

Adjoint operator

yields Whitney-Hodge decomposition

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57) by construction

deRham map DodziukPatodirsquo76

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of area

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Convergence of Hodge decomposition

Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)

bull This remains valid for variable (and converging) metrics

bull FEM-Whitney relations

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Thank you

• Algebraic Topology on Polyhedrafrom Linear Finite Elements
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Harmonic vector fields
• Contents
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Contents
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Contents
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Contents
• Discrete Harmonic vector fields
• Discrete Harmonic vector fields
• Computing harmonic fields
• Computing harmonic fields
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Discrete Hodge Star
• Discrete Hodge Star
• Discrete Hodge Star
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• Contents
• Convergence What can go wrong
• Convergence What can go wrong
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition

Computing harmonic fields

Harmonic fields cont

Harmonic fields cont

no tangential jump

X is closed

Harmonic fields cont

no tangential jump

X is co-closed

X is closed

Harmonic fields amp harmonic1-forms

(no tangential jump)

X is closed ie X corresponds tosimplicial 1-form

X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

this 1-form is co-closed(condition at vertices)X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

this 1-form is closed(condition at faces)X is pcw cnst

this 1-form is co-closed(condition at vertices)X is co-closed

this 1-form is co-closed(condition at vertices)

this 1-form is closed(condition at faces)X is pcw cnst harmonic 1-froms on graphs

cf GortlerGotsmanThurston

X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

(watch out cotan edge weightscould be negative cf BobenkoSpringborn forgetting non-negative weights by intrinsic edge flips)

Discrete Hodge Star

Complex multiplication (rotation by )

Discrete Hodge Star

X has no normal jump across edges

Complex multiplication (rotation by )

hellipdoes not take to itself

Because

X has no tangential jump across edges

X would have to be globally continuous

Discrete Hodge Star

Complex multiplication (rotation by )

Hodge star

hellipin other words Hodge star of X is harmonic part of J(X)

hellipdoes not take to itself

Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

(image are piecewise linear forms)

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

Adjoint operator

yields Whitney-Hodge decomposition

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57) by construction

deRham map DodziukPatodirsquo76

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of area

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Convergence of Hodge decomposition

Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)

bull This remains valid for variable (and converging) metrics

bull FEM-Whitney relations

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Thank you

• Algebraic Topology on Polyhedrafrom Linear Finite Elements
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Harmonic vector fields
• Contents
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Contents
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Contents
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Contents
• Discrete Harmonic vector fields
• Discrete Harmonic vector fields
• Computing harmonic fields
• Computing harmonic fields
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Discrete Hodge Star
• Discrete Hodge Star
• Discrete Hodge Star
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• Contents
• Convergence What can go wrong
• Convergence What can go wrong
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition

Harmonic fields cont

Harmonic fields cont

no tangential jump

X is closed

Harmonic fields cont

no tangential jump

X is co-closed

X is closed

Harmonic fields amp harmonic1-forms

(no tangential jump)

X is closed ie X corresponds tosimplicial 1-form

X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

this 1-form is co-closed(condition at vertices)X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

this 1-form is closed(condition at faces)X is pcw cnst

this 1-form is co-closed(condition at vertices)X is co-closed

this 1-form is co-closed(condition at vertices)

this 1-form is closed(condition at faces)X is pcw cnst harmonic 1-froms on graphs

cf GortlerGotsmanThurston

X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

(watch out cotan edge weightscould be negative cf BobenkoSpringborn forgetting non-negative weights by intrinsic edge flips)

Discrete Hodge Star

Complex multiplication (rotation by )

Discrete Hodge Star

X has no normal jump across edges

Complex multiplication (rotation by )

hellipdoes not take to itself

Because

X has no tangential jump across edges

X would have to be globally continuous

Discrete Hodge Star

Complex multiplication (rotation by )

Hodge star

hellipin other words Hodge star of X is harmonic part of J(X)

hellipdoes not take to itself

Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

(image are piecewise linear forms)

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

Adjoint operator

yields Whitney-Hodge decomposition

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57) by construction

deRham map DodziukPatodirsquo76

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of area

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Convergence of Hodge decomposition

Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)

bull This remains valid for variable (and converging) metrics

bull FEM-Whitney relations

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Thank you

• Algebraic Topology on Polyhedrafrom Linear Finite Elements
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Harmonic vector fields
• Contents
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Contents
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Contents
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Contents
• Discrete Harmonic vector fields
• Discrete Harmonic vector fields
• Computing harmonic fields
• Computing harmonic fields
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Discrete Hodge Star
• Discrete Hodge Star
• Discrete Hodge Star
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• Contents
• Convergence What can go wrong
• Convergence What can go wrong
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition

Harmonic fields cont

no tangential jump

X is closed

Harmonic fields cont

no tangential jump

X is co-closed

X is closed

Harmonic fields amp harmonic1-forms

(no tangential jump)

X is closed ie X corresponds tosimplicial 1-form

X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

this 1-form is co-closed(condition at vertices)X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

this 1-form is closed(condition at faces)X is pcw cnst

this 1-form is co-closed(condition at vertices)X is co-closed

this 1-form is co-closed(condition at vertices)

this 1-form is closed(condition at faces)X is pcw cnst harmonic 1-froms on graphs

cf GortlerGotsmanThurston

X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

(watch out cotan edge weightscould be negative cf BobenkoSpringborn forgetting non-negative weights by intrinsic edge flips)

Discrete Hodge Star

Complex multiplication (rotation by )

Discrete Hodge Star

X has no normal jump across edges

Complex multiplication (rotation by )

hellipdoes not take to itself

Because

X has no tangential jump across edges

X would have to be globally continuous

Discrete Hodge Star

Complex multiplication (rotation by )

Hodge star

hellipin other words Hodge star of X is harmonic part of J(X)

hellipdoes not take to itself

Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

(image are piecewise linear forms)

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

Adjoint operator

yields Whitney-Hodge decomposition

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57) by construction

deRham map DodziukPatodirsquo76

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of area

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Convergence of Hodge decomposition

Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)

bull This remains valid for variable (and converging) metrics

bull FEM-Whitney relations

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Thank you

• Algebraic Topology on Polyhedrafrom Linear Finite Elements
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Harmonic vector fields
• Contents
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Contents
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Contents
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Contents
• Discrete Harmonic vector fields
• Discrete Harmonic vector fields
• Computing harmonic fields
• Computing harmonic fields
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Discrete Hodge Star
• Discrete Hodge Star
• Discrete Hodge Star
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• Contents
• Convergence What can go wrong
• Convergence What can go wrong
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition

Harmonic fields cont

no tangential jump

X is co-closed

X is closed

Harmonic fields amp harmonic1-forms

(no tangential jump)

X is closed ie X corresponds tosimplicial 1-form

X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

this 1-form is co-closed(condition at vertices)X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

this 1-form is closed(condition at faces)X is pcw cnst

this 1-form is co-closed(condition at vertices)X is co-closed

this 1-form is co-closed(condition at vertices)

this 1-form is closed(condition at faces)X is pcw cnst harmonic 1-froms on graphs

cf GortlerGotsmanThurston

X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

(watch out cotan edge weightscould be negative cf BobenkoSpringborn forgetting non-negative weights by intrinsic edge flips)

Discrete Hodge Star

Complex multiplication (rotation by )

Discrete Hodge Star

X has no normal jump across edges

Complex multiplication (rotation by )

hellipdoes not take to itself

Because

X has no tangential jump across edges

X would have to be globally continuous

Discrete Hodge Star

Complex multiplication (rotation by )

Hodge star

hellipin other words Hodge star of X is harmonic part of J(X)

hellipdoes not take to itself

Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

(image are piecewise linear forms)

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

Adjoint operator

yields Whitney-Hodge decomposition

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57) by construction

deRham map DodziukPatodirsquo76

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of area

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Convergence of Hodge decomposition

Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)

bull This remains valid for variable (and converging) metrics

bull FEM-Whitney relations

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Thank you

• Algebraic Topology on Polyhedrafrom Linear Finite Elements
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Harmonic vector fields
• Contents
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Contents
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Contents
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Contents
• Discrete Harmonic vector fields
• Discrete Harmonic vector fields
• Computing harmonic fields
• Computing harmonic fields
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Discrete Hodge Star
• Discrete Hodge Star
• Discrete Hodge Star
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• Contents
• Convergence What can go wrong
• Convergence What can go wrong
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition

Harmonic fields amp harmonic1-forms

(no tangential jump)

X is closed ie X corresponds tosimplicial 1-form

X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

this 1-form is co-closed(condition at vertices)X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

this 1-form is closed(condition at faces)X is pcw cnst

this 1-form is co-closed(condition at vertices)X is co-closed

this 1-form is co-closed(condition at vertices)

this 1-form is closed(condition at faces)X is pcw cnst harmonic 1-froms on graphs

cf GortlerGotsmanThurston

X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

(watch out cotan edge weightscould be negative cf BobenkoSpringborn forgetting non-negative weights by intrinsic edge flips)

Discrete Hodge Star

Complex multiplication (rotation by )

Discrete Hodge Star

X has no normal jump across edges

Complex multiplication (rotation by )

hellipdoes not take to itself

Because

X has no tangential jump across edges

X would have to be globally continuous

Discrete Hodge Star

Complex multiplication (rotation by )

Hodge star

hellipin other words Hodge star of X is harmonic part of J(X)

hellipdoes not take to itself

Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

(image are piecewise linear forms)

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

Adjoint operator

yields Whitney-Hodge decomposition

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57) by construction

deRham map DodziukPatodirsquo76

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of area

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Convergence of Hodge decomposition

Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)

bull This remains valid for variable (and converging) metrics

bull FEM-Whitney relations

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Thank you

• Algebraic Topology on Polyhedrafrom Linear Finite Elements
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Harmonic vector fields
• Contents
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Contents
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Contents
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Contents
• Discrete Harmonic vector fields
• Discrete Harmonic vector fields
• Computing harmonic fields
• Computing harmonic fields
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Discrete Hodge Star
• Discrete Hodge Star
• Discrete Hodge Star
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• Contents
• Convergence What can go wrong
• Convergence What can go wrong
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

this 1-form is co-closed(condition at vertices)X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

this 1-form is closed(condition at faces)X is pcw cnst

this 1-form is co-closed(condition at vertices)X is co-closed

this 1-form is co-closed(condition at vertices)

this 1-form is closed(condition at faces)X is pcw cnst harmonic 1-froms on graphs

cf GortlerGotsmanThurston

X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

(watch out cotan edge weightscould be negative cf BobenkoSpringborn forgetting non-negative weights by intrinsic edge flips)

Discrete Hodge Star

Complex multiplication (rotation by )

Discrete Hodge Star

X has no normal jump across edges

Complex multiplication (rotation by )

hellipdoes not take to itself

Because

X has no tangential jump across edges

X would have to be globally continuous

Discrete Hodge Star

Complex multiplication (rotation by )

Hodge star

hellipin other words Hodge star of X is harmonic part of J(X)

hellipdoes not take to itself

Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

(image are piecewise linear forms)

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

Adjoint operator

yields Whitney-Hodge decomposition

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57) by construction

deRham map DodziukPatodirsquo76

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of area

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Convergence of Hodge decomposition

Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)

bull This remains valid for variable (and converging) metrics

bull FEM-Whitney relations

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Thank you

• Algebraic Topology on Polyhedrafrom Linear Finite Elements
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Harmonic vector fields
• Contents
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Contents
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Contents
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Contents
• Discrete Harmonic vector fields
• Discrete Harmonic vector fields
• Computing harmonic fields
• Computing harmonic fields
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Discrete Hodge Star
• Discrete Hodge Star
• Discrete Hodge Star
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• Contents
• Convergence What can go wrong
• Convergence What can go wrong
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

this 1-form is closed(condition at faces)X is pcw cnst

this 1-form is co-closed(condition at vertices)X is co-closed

this 1-form is co-closed(condition at vertices)

this 1-form is closed(condition at faces)X is pcw cnst harmonic 1-froms on graphs

cf GortlerGotsmanThurston

X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

(watch out cotan edge weightscould be negative cf BobenkoSpringborn forgetting non-negative weights by intrinsic edge flips)

Discrete Hodge Star

Complex multiplication (rotation by )

Discrete Hodge Star

X has no normal jump across edges

Complex multiplication (rotation by )

hellipdoes not take to itself

Because

X has no tangential jump across edges

X would have to be globally continuous

Discrete Hodge Star

Complex multiplication (rotation by )

Hodge star

hellipin other words Hodge star of X is harmonic part of J(X)

hellipdoes not take to itself

Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

(image are piecewise linear forms)

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

Adjoint operator

yields Whitney-Hodge decomposition

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57) by construction

deRham map DodziukPatodirsquo76

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of area

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Convergence of Hodge decomposition

Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)

bull This remains valid for variable (and converging) metrics

bull FEM-Whitney relations

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Thank you

• Algebraic Topology on Polyhedrafrom Linear Finite Elements
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Harmonic vector fields
• Contents
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Contents
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Contents
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Contents
• Discrete Harmonic vector fields
• Discrete Harmonic vector fields
• Computing harmonic fields
• Computing harmonic fields
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Discrete Hodge Star
• Discrete Hodge Star
• Discrete Hodge Star
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• Contents
• Convergence What can go wrong
• Convergence What can go wrong
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition

this 1-form is co-closed(condition at vertices)

this 1-form is closed(condition at faces)X is pcw cnst harmonic 1-froms on graphs

cf GortlerGotsmanThurston

X is co-closed

Harmonic fields amp harmonic1-forms

X is closed ie X corresponds tosimplicial 1-form

(watch out cotan edge weightscould be negative cf BobenkoSpringborn forgetting non-negative weights by intrinsic edge flips)

Discrete Hodge Star

Complex multiplication (rotation by )

Discrete Hodge Star

X has no normal jump across edges

Complex multiplication (rotation by )

hellipdoes not take to itself

Because

X has no tangential jump across edges

X would have to be globally continuous

Discrete Hodge Star

Complex multiplication (rotation by )

Hodge star

hellipin other words Hodge star of X is harmonic part of J(X)

hellipdoes not take to itself

Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

(image are piecewise linear forms)

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

Adjoint operator

yields Whitney-Hodge decomposition

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57) by construction

deRham map DodziukPatodirsquo76

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of area

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Convergence of Hodge decomposition

Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)

bull This remains valid for variable (and converging) metrics

bull FEM-Whitney relations

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Thank you

• Algebraic Topology on Polyhedrafrom Linear Finite Elements
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Harmonic vector fields
• Contents
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Contents
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Contents
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Contents
• Discrete Harmonic vector fields
• Discrete Harmonic vector fields
• Computing harmonic fields
• Computing harmonic fields
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Discrete Hodge Star
• Discrete Hodge Star
• Discrete Hodge Star
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• Contents
• Convergence What can go wrong
• Convergence What can go wrong
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition

Discrete Hodge Star

Complex multiplication (rotation by )

Discrete Hodge Star

X has no normal jump across edges

Complex multiplication (rotation by )

hellipdoes not take to itself

Because

X has no tangential jump across edges

X would have to be globally continuous

Discrete Hodge Star

Complex multiplication (rotation by )

Hodge star

hellipin other words Hodge star of X is harmonic part of J(X)

hellipdoes not take to itself

Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

(image are piecewise linear forms)

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

Adjoint operator

yields Whitney-Hodge decomposition

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57) by construction

deRham map DodziukPatodirsquo76

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of area

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Convergence of Hodge decomposition

Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)

bull This remains valid for variable (and converging) metrics

bull FEM-Whitney relations

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Thank you

• Algebraic Topology on Polyhedrafrom Linear Finite Elements
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Harmonic vector fields
• Contents
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Contents
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Contents
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Contents
• Discrete Harmonic vector fields
• Discrete Harmonic vector fields
• Computing harmonic fields
• Computing harmonic fields
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Discrete Hodge Star
• Discrete Hodge Star
• Discrete Hodge Star
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• Contents
• Convergence What can go wrong
• Convergence What can go wrong
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition

Discrete Hodge Star

X has no normal jump across edges

Complex multiplication (rotation by )

hellipdoes not take to itself

Because

X has no tangential jump across edges

X would have to be globally continuous

Discrete Hodge Star

Complex multiplication (rotation by )

Hodge star

hellipin other words Hodge star of X is harmonic part of J(X)

hellipdoes not take to itself

Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

(image are piecewise linear forms)

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

Adjoint operator

yields Whitney-Hodge decomposition

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57) by construction

deRham map DodziukPatodirsquo76

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of area

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Convergence of Hodge decomposition

Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)

bull This remains valid for variable (and converging) metrics

bull FEM-Whitney relations

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Thank you

• Algebraic Topology on Polyhedrafrom Linear Finite Elements
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Harmonic vector fields
• Contents
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Contents
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Contents
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Contents
• Discrete Harmonic vector fields
• Discrete Harmonic vector fields
• Computing harmonic fields
• Computing harmonic fields
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Discrete Hodge Star
• Discrete Hodge Star
• Discrete Hodge Star
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• Contents
• Convergence What can go wrong
• Convergence What can go wrong
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition

Discrete Hodge Star

Complex multiplication (rotation by )

Hodge star

hellipin other words Hodge star of X is harmonic part of J(X)

hellipdoes not take to itself

Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

(image are piecewise linear forms)

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

Adjoint operator

yields Whitney-Hodge decomposition

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57) by construction

deRham map DodziukPatodirsquo76

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of area

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Convergence of Hodge decomposition

Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)

bull This remains valid for variable (and converging) metrics

bull FEM-Whitney relations

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Thank you

• Algebraic Topology on Polyhedrafrom Linear Finite Elements
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Harmonic vector fields
• Contents
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Contents
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Contents
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Contents
• Discrete Harmonic vector fields
• Discrete Harmonic vector fields
• Computing harmonic fields
• Computing harmonic fields
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Discrete Hodge Star
• Discrete Hodge Star
• Discrete Hodge Star
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• Contents
• Convergence What can go wrong
• Convergence What can go wrong
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

(image are piecewise linear forms)

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

Adjoint operator

yields Whitney-Hodge decomposition

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57) by construction

deRham map DodziukPatodirsquo76

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of area

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Convergence of Hodge decomposition

Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)

bull This remains valid for variable (and converging) metrics

bull FEM-Whitney relations

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Thank you

• Algebraic Topology on Polyhedrafrom Linear Finite Elements
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Harmonic vector fields
• Contents
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Contents
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Contents
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Contents
• Discrete Harmonic vector fields
• Discrete Harmonic vector fields
• Computing harmonic fields
• Computing harmonic fields
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Discrete Hodge Star
• Discrete Hodge Star
• Discrete Hodge Star
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• Contents
• Convergence What can go wrong
• Convergence What can go wrong
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

Adjoint operator

yields Whitney-Hodge decomposition

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57) by construction

deRham map DodziukPatodirsquo76

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of area

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Convergence of Hodge decomposition

Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)

bull This remains valid for variable (and converging) metrics

bull FEM-Whitney relations

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Thank you

• Algebraic Topology on Polyhedrafrom Linear Finite Elements
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Harmonic vector fields
• Contents
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Contents
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Contents
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Contents
• Discrete Harmonic vector fields
• Discrete Harmonic vector fields
• Computing harmonic fields
• Computing harmonic fields
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Discrete Hodge Star
• Discrete Hodge Star
• Discrete Hodge Star
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• Contents
• Convergence What can go wrong
• Convergence What can go wrong
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57)

Inner product on simplicial co-chains

Adjoint operator

yields Whitney-Hodge decomposition

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57) by construction

deRham map DodziukPatodirsquo76

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of area

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Convergence of Hodge decomposition

Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)

bull This remains valid for variable (and converging) metrics

bull FEM-Whitney relations

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Thank you

• Algebraic Topology on Polyhedrafrom Linear Finite Elements
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Harmonic vector fields
• Contents
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Contents
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Contents
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Contents
• Discrete Harmonic vector fields
• Discrete Harmonic vector fields
• Computing harmonic fields
• Computing harmonic fields
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Discrete Hodge Star
• Discrete Hodge Star
• Discrete Hodge Star
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• Contents
• Convergence What can go wrong
• Convergence What can go wrong
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition

compare Whitney Forms

Simplicial co-chains

Whitney Forms (rsquo57) by construction

deRham map DodziukPatodirsquo76

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of area

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Convergence of Hodge decomposition

Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)

bull This remains valid for variable (and converging) metrics

bull FEM-Whitney relations

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Thank you

• Algebraic Topology on Polyhedrafrom Linear Finite Elements
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Harmonic vector fields
• Contents
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Contents
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Contents
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Contents
• Discrete Harmonic vector fields
• Discrete Harmonic vector fields
• Computing harmonic fields
• Computing harmonic fields
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Discrete Hodge Star
• Discrete Hodge Star
• Discrete Hodge Star
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• Contents
• Convergence What can go wrong
• Convergence What can go wrong
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition

Contents

Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of area

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Convergence of Hodge decomposition

Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)

bull This remains valid for variable (and converging) metrics

bull FEM-Whitney relations

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Thank you

• Algebraic Topology on Polyhedrafrom Linear Finite Elements
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Harmonic vector fields
• Contents
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Contents
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Contents
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Contents
• Discrete Harmonic vector fields
• Discrete Harmonic vector fields
• Computing harmonic fields
• Computing harmonic fields
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Discrete Hodge Star
• Discrete Hodge Star
• Discrete Hodge Star
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• Contents
• Convergence What can go wrong
• Convergence What can go wrong
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of area

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Convergence of Hodge decomposition

Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)

bull This remains valid for variable (and converging) metrics

bull FEM-Whitney relations

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Thank you

• Algebraic Topology on Polyhedrafrom Linear Finite Elements
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Harmonic vector fields
• Contents
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Contents
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Contents
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Contents
• Discrete Harmonic vector fields
• Discrete Harmonic vector fields
• Computing harmonic fields
• Computing harmonic fields
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Discrete Hodge Star
• Discrete Hodge Star
• Discrete Hodge Star
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• Contents
• Convergence What can go wrong
• Convergence What can go wrong
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition

Convergence What can go wrong

Lantern of Schwarz

hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Convergence of Hodge decomposition

Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)

bull This remains valid for variable (and converging) metrics

bull FEM-Whitney relations

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Thank you

• Algebraic Topology on Polyhedrafrom Linear Finite Elements
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Harmonic vector fields
• Contents
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Contents
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Contents
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Contents
• Discrete Harmonic vector fields
• Discrete Harmonic vector fields
• Computing harmonic fields
• Computing harmonic fields
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Discrete Hodge Star
• Discrete Hodge Star
• Discrete Hodge Star
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• Contents
• Convergence What can go wrong
• Convergence What can go wrong
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Convergence of Hodge decomposition

Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)

bull This remains valid for variable (and converging) metrics

bull FEM-Whitney relations

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Thank you

• Algebraic Topology on Polyhedrafrom Linear Finite Elements
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Harmonic vector fields
• Contents
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Contents
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Contents
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Contents
• Discrete Harmonic vector fields
• Discrete Harmonic vector fields
• Computing harmonic fields
• Computing harmonic fields
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Discrete Hodge Star
• Discrete Hodge Star
• Discrete Hodge Star
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• Contents
• Convergence What can go wrong
• Convergence What can go wrong
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition

Convergence of Hodge decomposition

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Convergence of Hodge decomposition

Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)

bull This remains valid for variable (and converging) metrics

bull FEM-Whitney relations

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Thank you

• Algebraic Topology on Polyhedrafrom Linear Finite Elements
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Harmonic vector fields
• Contents
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Contents
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Contents
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Contents
• Discrete Harmonic vector fields
• Discrete Harmonic vector fields
• Computing harmonic fields
• Computing harmonic fields
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Discrete Hodge Star
• Discrete Hodge Star
• Discrete Hodge Star
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• Contents
• Convergence What can go wrong
• Convergence What can go wrong
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition

Convergence of Hodge decomposition

Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)

bull This remains valid for variable (and converging) metrics

bull FEM-Whitney relations

-projection of smooth to piecewise constant vector fields

discrete Hodge

smooth Hodge

Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star

Thank you

• Algebraic Topology on Polyhedrafrom Linear Finite Elements
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Harmonic vector fields
• Contents
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Contents
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Contents
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Contents
• Discrete Harmonic vector fields
• Discrete Harmonic vector fields
• Computing harmonic fields
• Computing harmonic fields
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Discrete Hodge Star
• Discrete Hodge Star
• Discrete Hodge Star
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• Contents
• Convergence What can go wrong
• Convergence What can go wrong
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition

Thank you

• Algebraic Topology on Polyhedrafrom Linear Finite Elements
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Letrsquos start with a concrete problem
• Harmonic vector fields
• Contents
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review smooth deRham complex
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Review deRham cohomology
• Contents
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Discrete deRham
• Contents
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Discrete Hodge for closed meshes
• Contents
• Discrete Harmonic vector fields
• Discrete Harmonic vector fields
• Computing harmonic fields
• Computing harmonic fields
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields cont
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Harmonic fields amp harmonic1-forms
• Discrete Hodge Star
• Discrete Hodge Star
• Discrete Hodge Star
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• compare Whitney Forms
• Contents
• Convergence What can go wrong
• Convergence What can go wrong
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition
• Convergence of Hodge decomposition