h(curl) and h(div) elements on polytopes from generalized

H(curl) and H(div) Elements on Polytopes fromGeneralized Barycentric Coordinates

Andrew Gillette

Department of MathematicsUniversity of Arizona

joint work with

Alex Rand (CD-adapco)Chandrajit Bajaj (UT Austin)

Andrew Gillette - U. Arizona ( )Vector Elements on Polytopes with GBCs Finite Element Circus 2014 1 / 13

The generalized barycentric coordinate approach

Let P be a convex polytope with vertex set V . We say that

λv : P → R are generalized barycentric coordinates (GBCs) on P

if they satisfy λv ≥ 0 on P and L =∑v∈V

L(vv)λv, ∀ L : P → R linear.

Familiar properties are implied by this definition:∑v∈V

λv ≡ 1︸︷︷︸partition of unity

∑v∈V

vλv(x) = x︸︷︷︸linear precision

λvi (vj ) = δij︸︷︷︸interpolation

traditional FEM family of GBC reference elements

Bilinear Map

Physical

Element

Reference

Element

Affine Map TUnit

Diameter

TΩΩ


Developments in GBC FEM theory1 Characterization of the dependence of error estimates on polytope geometry.

vs.

2 Construction of higher order scalar-valued methods using λv functions.

λi λiλj ψij

3 Construction of H(curl) and H(div) methods using λv and ∇λv functions.

H1 grad // H(curl) curl // H(div)div // L2

λi λi∇λj λi∇λj ×∇λk χP


Many choices of generalized barycentric coordinates

Triangulation⇒ FLOATER, HORMANN, KÓS, A generalconstruction of barycentric coordinatesover convex polygons, 2006

0 ≤ λTmi (x) ≤ λi (x) ≤ λTM

i (x) ≤ 1

Wachspress⇒ WACHSPRESS, A Rational FiniteElement Basis, 1975.⇒ WARREN, Barycentric coordinates forconvex polytopes, 1996.

Sibson / Laplace⇒ SIBSON, A vector identity for theDirichlet tessellation, 1980.⇒ HIYOSHI, SUGIHARA, Voronoi-basedinterpolation with higher continuity, 2000.


Many choices of generalized barycentric coordinates

x

vi

vi+1

vi−1

αi−1

αi ri

∆u = 0

Mean value⇒ FLOATER, Mean value coordinates, 2003.⇒ FLOATER, KÓS, REIMERS, Mean value coordinates in3D, 2005.

Harmonic⇒ WARREN, SCHAEFER, HIRANI, DESBRUN, Barycentriccoordinates for convex sets, 2007.⇒ CHRISTIANSEN, A construction of spaces of compatibledifferential forms on cellular complexes, 2008.

Many more papers could be cited (maximum entropy coordinates, moving leastsquares coordinates, surface barycentric coordinates, etc...)


From scalar to vector elements

The classical finite element sequences for a domain Ω ⊂ Rn are written:

n = 2 : H1 grad // H(curl) oo rot // H(div)div // L2

n = 3 : H1 grad // H(curl) curl // H(div)div // L2

These correspond to the L2 deRham diagrams from differential topology:

n = 2 : HΛ0 d0 // HΛ1 ∼= // HΛ1oo d1 // HΛ2

n = 3 : HΛ0 d0 // HΛ1 d1 // HΛ2 d2 // HΛ3

Conforming finite element subspaces of HΛk are of two types:

Pr Λk := k -forms with degree r polynomial coefficients

P−r Λk := Pr−1Λk ⊕ certain additional k -forms

This notation, from Finite Element Exterior Calculus, can be used to describe manywell-known finite element spaces.

ARNOLD, FALK, WINTHER Finite Element Exterior Calculus, Bulletin of the AMS, 2010.


Classical finite element spaces on simplices

n=2 (triangles)

k dim space type classical description0 3 P1Λ0 H1 Lagrange elements of degree ≤ 1

3 P−1 Λ0 H1 Lagrange elements of degree ≤ 11 6 P1Λ1 H(div) Brezzi-Douglas-Marini H(div) elements of degree ≤ 1

3 P−1 Λ1 H(div) Raviart-Thomas elements of order 02 3 P1Λ2 L2 discontinuous linear

1 P−1 Λ2 L2 discontinuous piecewise constant

n=3 (tetrahedra)

0 4 P1Λ0 H1 Lagrange elements of degree ≤ 14 P−1 Λ0 H1 Lagrange elements of degree ≤ 1

1 12 P1Λ1 H(curl) Nédélec second kind H(curl) elements of degree ≤ 16 P−1 Λ1 H(curl) Nédélec first kind H(curl) elements of order 0

2 12 P1Λ2 H(div) Nédélec second kind H(div) elements of degree ≤ 14 P−1 Λ2 H(div) Nédélec first kind H(div) elements of order 0

3 4 P1Λ3 L2 discontinuous linear1 P−1 Λ3 L2 discontinuous piecewise constant


Basis functions on simplices

n=2 (triangles)

k dim space type basis functions0 3 P1Λ0 H1 λi

1 6 P1Λ1 H(curl) λi∇λj

6 P1Λ1 H(div) rot(λi∇λj )

3 P−1 Λ1 H(curl) λi∇λj − λj∇λi

3 P−1 Λ1 H(div) rot(λi∇λj − λj∇λi )

2 3 P1Λ2 L2 piecewise linear functions1 P−1 Λ2 L2 piecewise constant functions

n=3 (tetrahedra)

0 4 P1Λ0 H1 λi

1 12 P1Λ1 H(curl) λi∇λj

6 P−1 Λ1 H(curl) λi∇λj − λj∇λi

2 12 P1Λ2 H(div) λi∇λj ×∇λk

4 P−1 Λ2 H(div) (λi∇λj ×∇λk ) + (λj∇λk ×∇λi ) + (λk∇λi ×∇λj )

3 4 P1Λ3 L2 piecewise linear functions1 P−1 Λ3 L2 piecewise constant functions


Essential properties of basis functionsThe vector-valued basis constructions (0 < k < n) have two key properties:

1 Global continuity in H(curl) or H(div)

λi∇λj agree on tangentialcomponents at element interfaces =⇒ H(curl) continuity

λi∇λj ×∇λk agree on normalcomponents at element interfaces =⇒ H(div) continuity

2 Reproduction of requisite polynomial differential forms.

For i, j ∈ 1, 2, 3:

spanλi∇λj = span[

10

],

[01

],

[x0

],

[0x

],

[y0

],

[0y

]∼= P1Λ1(R2)

spanλi∇λj − λj∇λi = span[

10

],

[01

],

[xy

]∼= P−1 Λ1(R2)

Using generalized barycentric coordinates, we can extend all these results topolygonal and polyhedral elements.


Basis functions on polygons and polyhedra

Theorem [G., Rand, Bajaj, 2014]Let P be a convex polygon or polyhedron. Given any set of generalized barycentriccoordinates λi associated to P, the functions listed below have global continuityand polynomial differential form reproduction properties as indicated.

n=2(polygons)

k space type functions

1 P1Λ1 H(curl) λi∇λj

P1Λ1 H(div) rot(λi∇λj )

P−1 Λ1 H(curl) λi∇λj − λj∇λi

P−1 Λ1 H(div) rot(λi∇λj − λj∇λi )

n=3(polyhedra)

1 P1Λ1 H(curl) λi∇λj

P−1 Λ1 H(curl) λi∇λj − λj∇λi

2 P1Λ2 H(div) λi∇λj ×∇λk

P−1 Λ2 H(div) (λi∇λj ×∇λk ) + (λj∇λk ×∇λi )+ (λk∇λi ×∇λj )

Note: The indices range over all pairs or triples of vertex indices from P.Andrew Gillette - U. Arizona ( )Vector Elements on Polytopes with GBCs Finite Element Circus 2014 10 / 13

Polynomial differential form reproduction identities

Let P ⊂ R3 be a convex polyhedron with vertex set vi. Let x =[x y z

]T .

Then for any 3× 3 real matrix A, ∑i,j

λi∇λj (vj − vi )T = I∑

i,j

(Avi · vj )(λi∇λj ) = Ax

12

∑i,j,k

λi∇λj ×∇λk ((vj − vi )× (vk − vi ))T = I

12

∑i,j,k

(Avi · (vj × vk ))(λi∇λj ×∇λk ) = Ax.

By appropriate choice of constant entries for A, the column vectors of I and Ax spanP1Λ1 ⊂ H(curl) or P1Λ2 ⊂ H(div).

→ Additional identities for the remaining cases are stated in:G, RAND, BAJAJ Construction of Scalar and Vector Finite Element Families onPolygonal and Polyhedral Meshes, arXiv:1405.6978, 2014


Reducing the basisIn some cases, it should be possible to reduce the size of the basis constructed by ourmethod, in an analogous fashion to the quadratic scalar case.

n=2 (polygons)

k space # construction # boundary # polynomial

0 P1Λ0(m)/P−0 Λ1(m) v v 3

1 P1Λ1(m) v(v − 1) 2e 6

P−1 Λ1(m)

(v2

)e 3

2 P1Λ2(m)v(v − 1)(v − 2)

20 3

P−1 Λ2(m)

(v3

)0 1

→ The n = 3 (polyhedra) version of this table is given in:

G, RAND, BAJAJ Construction of Scalar and Vector Finite Element Families onPolygonal and Polyhedral Meshes, arXiv:1405.6978, 2014


Acknowledgments

Chandrajit Bajaj UT Austin

Alexander Rand UT Austin / CD-adapco

Happy birthday, Doug!

Slides and pre-prints: http://math.arizona.edu/~agillette/

More on GBCs: http://www.inf.usi.ch/hormann/barycentric


http://math.arizona.edu/~agillette/

http://www.inf.usi.ch/hormann/barycentric

h(curl) and h(div) elements on polytopes from generalized

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