adaptive finite elements with large aspect ratio: theory ... · adaptive nite elements with large...

Adaptive finite elements with large aspect ratio:theory and practice

M. Picasso

Institut d’Analyse et Calcul ScientifiqueEcole Polytechnique Federale de Lausanne

and

Projet GammaINRIA Rocquencourt

INRIA Rocquencourt, january 2010

M. Picasso Adaptive finite elements with large aspect ratio

Adaptive finite elements

Theory + practice: a posteriori error estimates + adaptivealgorithms → mathematicians can have the lead.

A posteriori error estimates for elliptic, parabolic, hyperbolic ?nonlinear ? problems + efficient meshing tools (INRIA, LJLL)→ can be used for industrial problems.

Adaptive finite element with large aspect ratio: the ultimatetool to reduce the number of vertices given a prescribed levelof accuracy.

Remark: Sparse tensor product methods are anisotropicmethods (shape functions).


Finite elements with large aspect ratio

General statement (mathematician): If nothing is knownabout the solution, then finite elements with large aspect ratioshould not be used.

A priori error estimates: ‖∇(u − uh)‖L2(Ω) ≤ Ch|u|H2(Ω) andC is large when the aspect ratio is large.

But engineers use finite elements with large aspect ratio.Example (Dassault Aviation): viscous compressible flowsaround aircrafts, aspect ratio 103.

More precise statement: finite elements with large aspect ratiocan be used provided the mesh fits the solution.

This is precisely the goal of adaptive finite elements with largeaspect ratio.

The theory of finite elements has to be revisited to handlemeshes with large aspect ratio.


Examples

Advection-diffusion (academic problem).

Flows around aircrafts (no theory, allows very fastcomputations).

Microfluidics (parabolic problem, theory and practice).


Example 1: 2D advection-diffusion

Mesh Isovalues

−ε∆u +~a · ∇u = f , boundary layer 10−4, 241 vertices, asp.ratio O(105), same precision with isotropic mesh O(105)vertices, Picasso SISC 2003.

Design of the stabilization parameter: anisotropic a priorierror estimates, Micheletti Perotto Picasso SINUM 2003.


Example 2: 3D flows around aircrafts

Bourgault Picasso Alauzet Loseille IJNMF 2009, supported byDassault Aviation.

Compressible Euler solver (Alauzet), Anisotropic meshgenerator (Dobrzynski Frey), Anisotropic error estimator.

Animation, Ma = 1.6, AoA = 3o , 106 vertices, single 32bprocessor.

Goal oriented: see A. Loseille’s PhD.


Example 2: 3D flows around aircrafts

With W. Hassan, F. Alauzet, supported by Dassault Aviation.

Compressible Navier-Stokes solver (Alauzet), Anisotropic meshgenerator (Dobrzynski Frey), Anisotropic error estimator.


Example 3: Microfluidics

Heat equation, Lozinski Picasso Prachittham SISC 2009.

Unsteady convection-diffusion, Picasso Prachittham JCAM2009.

Microfluidics, Picasso Prachittham Gijs IJNMF 2009.

Adaptive time steps and finite elements with large aspectratio.

Optimal a posteriori error estimates for the Crank Nicolsonscheme.

Animation.

Animation.


A priori and a posteriori error estimates

A prioriIsotropic meshes: ∃C > 0 (dep. aspect ratio, indep. u, h)

‖∇(u − uh)‖L2(Ω) ≤ Ch|u|H2(Ω).

Anisotropic meshes: Hessian matrix H(u).Frey George 2008, Chen Sun Xu 2007, Mirebeau Cohen 2009.Formaggia Perotto 2001, ∃C > 0 (indep. aspect ratio, u, h)Z

K

|∇(u − rhu)|2 ≤ C

λ4

1,K

λ22,K

ZK

(~rT1,KH(u)~r1,K )2

+ 2λ21,K

ZK

(~rT1,KH(u)~r2,K )2 + λ2

2,K

ZK

(~rT2,KH(u)~r2,K )2

!.

Ex: the mesh is aligned with u, u = u(x2), ~r1,K = (1 0)TZK

|∇(u − rhu)|2 ≤ Cλ22,K

ZK

„∂2u

∂x22

«2

.

Most anisotropic adaptive algorithms use a priori errorestimates and an estimate of H(u).

A posteriori


A priori and a posteriori error estimates

A priori

A posteriori

Isotropic meshes: ∃C1,C2 > 0 (dep. aspect ratio, indep. u, h)

C1η ≤ ‖∇(u − uh)‖L2(Ω) ≤ C2η + h.o.t.,

where the error estimator η is a computable quantity dep. onthe mesh size, the data and on uh.Anisotropic meshes:

Kunert 2000: C2 depends on the alignement of the mesh withthe (unknown) solution u.Picasso 2003, 2006: C1 and C2 indep. aspect ratio, u, h,provided the error estimator is equidistributed in the directionsof min. and max. stretching.


Outline

The anisotropic error estimator for the Laplace equation.

Adaptive algorithms.

Extension to parabolic problems.


The anisotropic error estimator for the Laplace problem

Find u : Ω→ R such that

−∆u = f in Ω,

u = 0 on ∂Ω.

Let Th be a mesh of Ω into triangles K with diameter hK lessthan h.

Find uh ∈ Vh (continuous, piecewise linears) such that, for allvh ∈ Vh ∫

Ω∇uh · ∇vh =

∫Ω

fvh.


A posteriori H1 error estimates - explicit, residual basederror estimator

The classical procedure to obtain an explicit, residual basederror estimator is:∫

Ω|∇(u − uh)|2 =

∫Ω

f (u − uh)−∫

Ω∇uh · ∇(u − uh),

=

∫Ω

f (u − uh − vh)−∫

Ω∇uh · ∇(u − uh − vh) ∀vh ∈ Vh,

=∑K∈Th

(∫K

(f + ∆uh)(u − uh − vh)

+1

2

∫∂K

[∇uh · n](u − uh − vh)

).

Use Cauchy-Schwarz inequality, take vh = Rh(u − uh)Clement interpolant, use anisotropic interpolation estimates toobtain ...


A posteriori H1 error estimates - explicit, residual basederror estimator

∫Ω|∇(u − uh)|2 ≤ C

∑K∈Th

η2K ,

where C does not depend on the aspect ratio.Here η2

K =(‖f + ∆uh‖L2(K) +

1

2

(|∂K |

λ1,Kλ2,K

)1/2

‖[∇uh · n]‖L2(∂K)

)(λ2

1,K

(~rT

1,KGK (u−uh)~r1,K

)+λ2

2,K

(~rT

2,KGK (u−uh)~r2,K

))1/2

,

with GK (v) =

∫

∆K

(∂v

∂x1

)2 ∫∆K

∂v

∂x1

∂v

∂x2∫∆K

∂v

∂x1

∂v

∂x2

∫∆K

(∂v

∂x2

)2

.


A posteriori H1 error estimates - anisotropic case

η2K =

(‖f + ∆uh‖L2(K) +

1

2

(|∂K |

λ1,Kλ2,K

)1/2

‖[∇uh · n]‖L2(∂K)

)(λ2

1,K

(~rT

1,KGK (u − uh)~r1,K

)+ λ2

2,K

(~rT


))1/2

.

What are λ1,K , λ2,K , ~r1,K , ~r2,K ?

Lower bound?

How to estimate G (u − uh)?

Isotropic case λ1,K ' λ2,K ' hK then the isotropic, explicit, residualbased error estimator is recovered.

Uniform stretching in the x1 direction, u(x2), uh(x2), then λ1,K canbe arbitrary large.

x1

x2

λ1,K

λ2,K

~r1,KK


Anisotropic interpolation estimates

Formaggia Perotto, Numer. Math. 2001, 2003.

See also Kunert, Kunert Verfurth, Numer. Math. 2000.

x1

x2

x1

x2

1

1

H

h

~r1,K

~r2,K

TK

K K

~x = TK (~x) = MK ~x +~tK , s. v. d. MK = RTK ΛKPK

RK =

(~rT

1,K

~rT2,K

)=

(1 00 1

), ΛK =

(λ1,K 0

0 λ2,K

)=

(H 00 h

).





x1

x2 TK

1K

K

~r1,K

~r2,K

λ1,K

λ2,K

~x = TK (~x) = MK ~x +~tK , s. v. d. MK = RTK ΛKPK

The unit circle ~xT ~x = 1 is mapped into the ellipse(~x −~tK )TRT

K Λ−2K RK (~x −~tK ) = 1.





The number of neighbours must be bounded above.

The reference patch T−1K (∆K ) must be O(1). This excludes

meshes having “large curvature”. In practice, anisotropicmesh generators satisfy this condition.

No conditions about the angles.


Acceptable patch: T−1K (∆K ) = O(1)

TK

K K

∆K

H1

1 h


Non acceptable patch: T−1K (∆K ) 6= O(1)

TK

K K

H1

1 h


Clement interpolant - anisotropic interpolation estimates



Clement interpolation estimates : ∃C > 0 indep. mesh sizeand aspect ratio s.t. ∀v ∈ H1(Ω), ∀K ∈ Th :

‖v − Rhv‖2L2(K) +

λ1,Kλ2,K

|∂K |‖v − Rhv‖2

L2(∂K)

≤ C

(λ2

1,K

(~rT

1,KGK (v)~r1,K

)+ λ2

2,K

(~rT

2,KGK (v)~r2,K

)),

GK (v) =

∫

∆K

(∂v

∂x1

)2

dx

∫∆K

∂v

∂x1

∂v

∂x2dx∫

∆K

∂v

∂x1

∂v

∂x2dx

∫∆K

(∂v

∂x2

)2

dx

.


A posteriori H1 error estimates - anisotropic case

η2K =

(‖f + ∆uh‖L2(K) +

1

2

(|∂K |

λ1,Kλ2,K

)1/2

‖[∇uh · n]‖L2(∂K)

)(λ2

1,K

(~rT


)+ λ2

2,K

(~rT


))1/2

.

Lower bound?

How to estimate G (u − uh)?


Lower bound?

η2K =

(‖f + ∆uh‖L2(K) +

1

2

(|∂K |

λ1,Kλ2,K

)1/2

‖[∇uh · n]‖L2(∂K)

)(λ2

1,K

(~rT


)+ λ2

2,K

(~rT


))1/2

.

Mimicking the proof of Verfurth (bubbles): if the mesh is such that∃C1 > 0 (indep. mesh size and aspect ratio) s.t.

λ21,K

(~rT

1,KGK (u−uh)~r1,K

)≤ C1 λ

22,K

(~rT

2,KGK (u−uh)~r2,K

)∀K ∈ Th

then ∃C2 > 0 (indep. mesh size and aspect ratio) s.t.

estimator ≤ C2 error + h.o.t.

Suggests to use an adaptive algorithm such that

λ21,K

(~rT

1,KGK (e)~r1,K

)' λ2

2,K

(~rT

2,KGK (e)~r2,K

)∀K ∈ Th


How to estimate G (u − uh)? Zienkiewicz-Zhu ZZ(post-processing)

From ∇uh, compute a better gradient → Πh∇uh (Πh is anapproximation of the L2 projection of ∇uh onto Vh).

Then ηZZ =

(∫Ω|∇uh − Πh∇uh|2

)1/2

is asymptotically exact

on parallel meshes (Rodriguez 1994, Ainsworth Oden 1997),

mildly structured meshes (Xu Zhang 2003), equivalent to thetrue error on general unstructured meshes (Carstensen 2004).

Superconvergence

‖∇u−∇uh‖L2(Ω) ≤ ‖∇u−Πh∇uh‖L2(Ω)+‖Πh∇uh−∇uh‖L2(Ω).


How to estimate G (u − uh) ? Zienkiewicz-Zhu ZZ(post-processing)

h1− h2 (1 : 1) error ei0.01− 0.01 1.36 0.81

0.005− 0.005 0.69 0.920.0025− 0.0025 0.35 0.97

h1− h2 (1 : 8) error ei0.005− 0.04 0.65 0.94

0.0025− 0.02 0.33 0.980.00125− 0.01 0.16 0.99

Open question: why ZZ so good on general meshes?


How to estimate G (u − uh) ? Zienkiewicz-Zhu ZZ(post-processing)

η2K =

(‖f + ∆uh‖L2(K) +

1

2

(|∂K |

λ1,Kλ2,K

)1/2

‖[∇uh · n]‖L2(∂K)

)(λ2

1,K

(~rT


)+ λ2

2,K

(~rT


))1/2

.

See also Kunert Nicaise M2AN 2003.

Zienkiewicz-Zhu∫K

(∂(u − uh)

∂x1

)2

→∫

K

(Πh∂uh

∂x1− ∂uh

∂x1

)2

.

Question: could we use only ZZ ?

η2K = ~rT

1,KGK (u − uh)~r1,K +~rT2,KGK (u − uh)~r2,K


Anisotropic, adaptive finite elements

Goal: find Th s.t. 0.75 TOL ≤

(∑K∈Th

η2K

)1/2

(∫Ω|∇uh|2

)1/2≤ 1.25 TOL

Sufficient condition (NK is the number of triangles):0.752TOL2

∫Ω|∇uh|2

NK≤ η2

K ≤1.252TOL2

∫Ω|∇uh|2

NK

Sufficient condition in the directions of maximum (i = 1) andminimum (i = 2) stretching:

0.752TOL2∫

Ω|∇uh|2√

2NK

≤

(‖f ‖L2(K) +

1

2

(|∂K |

λ1,Kλ2,K

)1/2

‖ [∇uh · ~n] ‖L2(∂K)

)(λ2

i,K

(~rTi,KGK (e)~ri,K

))1/2

≤1.252TOL2

∫Ω|∇uh|2√

2NK


Anisotropic, adaptive finite elements

P

h1,P

h2,P

θP

Equidistribute the error in directions 1 and 2

Align the triangle with the eigenvectors of GK (u − uh)

2D: use the BL2D mesh generator (INRIA, Borouchaki, Laug)or the BAMG mesh generator (Hecht).

3D: use the MeshAdapt mesh generator (INRIA, George HechtSaltel, Distene) or the MMG3D mesh generator (DobrzynskiFrey).


Adaptive meshes for the Laplace problem in 2D

2D: TOL = 0.25, 30 mesh generations, animation, zoom.

The effectivity index is aspect ratio independent on adaptedmeshes

TOL vertices error ei eiZZ asp. ratio

0.125 854 0.25 2.70 1.00 2620.0625 2793 0.13 2.75 0.99 288

0.03125 10812 0.062 2.79 0.95 4250.015625 42562 0.031 2.79 0.98 1199

3D: TOL = 0.25, 30 mesh generations, animation, zoom.


The heat equation with space discretization only

∂u

∂t−∆u = f in Ω× (0,T ).

Find uh : t → uh(·, t) ∈ Vh such that, for all t ∈ (0,T ), for allvh ∈ Vh ∫

Ω

∂uh

∂tvhdx +

∫Ω∇uh · ∇vhdx =

∫Ω

fvhdx .

e = u − uh⟨∂e

∂t, e

⟩+

∫Ω|∇e|2dx =

∫Ω

(f − ∂uh

∂t

)e dx−

∫Ω∇uh·∇edx

=

∫Ω

(f − ∂uh

∂t

)(e − vh) dx −

∫Ω∇uh · ∇(e − vh)dx

Use Cauchy-Schwarz inequality, take vh = Rhe Clementinterpolant, use anisotropic interpolation estimates to obtain...


The heat equation with space discretization only

1

2

∫Ω

(u − uh)2(T )dx +

∫ T

0

∫Ω|∇(u − uh)(t)|2dxdt

≤ 1

2

∫Ω

(u − uh)2(0)dx + C

∫ T

0

∑K∈Th

η2K ,

where C does not depend on the mesh size and aspect ratio.

Here η2K =(

‖f − ∂uh

∂t+ ∆uh‖L2(K) +

1

2

(|∂K |

λ1,Kλ2,K

)1/2

‖[∇uh · n]‖L2(∂K)

)(λ2

1,K

(~rT

1,KGK (u−uh)~r1,K

)+λ2

2,K

(~rT

2,KGK (u−uh)~r2,K

))1/2

.


The heat equation with Euler backward scheme

∂u

∂t−∆u = f in Ω× (0,T ).

For n = 1, ...,N, find unh ∈ Vh such that, for all vh ∈ Vh

1

τ

∫Ω

(unh − un−1

h )vhdx +

∫Ω∇un

h · ∇vhdx =

∫Ω

f nvhvdx .

Isotropic meshes: Picasso 1998, Verfurth 2003, BergamBernardi Mghazli 2004.

uhτ (x , t) =t − tn−1

τunh(x) +

tn − t

τun−1h (x).∫

Ω

∂uhτ

∂tvhdx +

∫Ω∇uhτ · ∇vhdx

=

∫Ω

fvhdx +

∫Ω

(f n − f )vhdx + (tn − t)

∫Ω∇∂uhτ

∂t· ∇vhdx .



e = u − uhτ⟨∂e

∂t, e

⟩+

∫Ω|∇e|2dx

=

∫Ω

(f − ∂uhτ

∂t

)e dx −

∫Ω∇uhτ · ∇edx

=

∫Ω

(f − ∂uhτ

∂t

)(e − vh) dx −

∫Ω∇uhτ · ∇(e − vh)dx

+

∫Ω

(f − f n)vhdx + (tn − t)

∫Ω∇∂uhτ

∂t· ∇vhdx .

Take vh = Rhe, use anisotropic interpolation estimates:

1

2

∫Ω

(u − uhτ )2(T )dx +

∫ T

0

∫Ω|∇(u − uhτ )(t)|2dxdt

≤ 1

2

∫Ω

(u − uhτ )2(0)dx + CN∑

n=1

∑K∈Th

η2K ,n.



with C independent of u, the mesh size and aspect ratio and

η2K ,n

=

∫ tn

tn−1

(∥∥∥∥f − ∂uhτ

∂t+ ∆uhτ

∥∥∥∥L2(K)

+1

2λ1/22,K

‖[∇uhτ · ~n]‖L2(∂K)

)(λ2

1,K

(~rT

1,KGK (u − uhτ )~r1,K

)+ λ2

2,K

(~rT


))1/2

+ ‖f − f n‖2L2(K) + τ2

n

∥∥∥∥∇∂uhτ

∂t

∥∥∥∥2

L2(K)

dt.


The heat equation with Crank-Nicolson scheme

Optimal L2(H1) a posteriori error estimates forCrank-Nicolson time discretization: Akrivis MakridadkisNochetto, Math. Comp. 2006.

For n = 1, ...,N, find unh ∈ Vh such that, for all vh ∈ Vh

1

τ

∫Ω

(unh−un−1

h )vhdx+1

2

∫Ω∇(un−1

h +unh)·∇vhdx =

1

2

∫Ω

(f n−1+f n)vhdx .

uhτ (x , t) =t − tn−1

τunh(x) +

tn − t

τun−1h (x).∫

Ω

∂uhτ

∂tvhdx +

∫Ω∇uhτ · ∇vhdx

=1

2

∫Ω

(f n−1 + f n)vhdx + (t − tn−1/2)

∫Ω∇∂uhτ

∂t· ∇vhdx .


The three points time reconstruction

Lozinski Prachittham Picasso SISC 2009.

Lagrange interpolant of degree 2: for tn−1 ≤ t ≤ tn

uhτ (x , t) =(t − tn−2)(t − tn−1)

(tn − tn−2)(tn − tn−1)unh

+(t − tn)(t − tn−2)

(tn−1 − tn)(tn−1 − tn−2)un−1h

+(t − tn)(t − tn−1)

(tn−2 − tn)(tn−2 − tn−1)un−2h .

Or equivalently

uhτ (x , t) = uhτ (x , t) +1

2(t − tn−1)(t − tn)∂2

nuh,

∂2nuh =

unh − un−1

h

τ−

un−1h − un−2

h

ττ

.


Error indicator

Approach

∫ T

t1

∫Ω

|∇(u − uhτ )|2 byN∑

n=2

∑K∈Th

η2K ,n with η2

K ,n =

∫ tn

tn−1

(∥∥∥∥f − ∂uhτ

∂t+ ∆uhτ

∥∥∥∥L2(K)

+1

2λ1/22,K

‖[∇uhτ · ~n]‖L2(∂K)

)(λ2

1,K

(~rT


)+ λ2

2,K

(~rT


))1/2

+

∥∥∥∥f − (f n−1/2 + (t − tn−1/2)f n − f n−2

2τ

)∥∥∥∥2

L2(K)

+τ 4∥∥∇∂2

nuh

∥∥2

L2(K)

dt.


Adaptive space-time algorithm

Choose the time step and the mesh size so that

0.875TOL ≤ ηspace + ηtime(∫ T

0

∫Ω|∇uhτ |2 dx dt

)1/2≤ 1.125TOL.

Test case 1 Animation

Test case 2 Animation

TOL error eiZZ ei space ei time Vert N Gen. AR0.125 0.03 0.99 2.87 2.68 155 142 84 63

0.0625 0.015 0.99 2.89 2.91 348 201 52 1080.03125 0.0078 0.99 2.96 2.99 892 285 52 165

0.015625 0.0040 1.00 2.88 2.71 4408 401 40 118

Conclusion: Vert = O(TOL−2) and N = O(TOL−1/2).


Conclusions

Adaptive finite elements with large aspect ratio → reduce thenumber of vertices.

Anisotropic interpolation estimates (Formaggia-Perotto,Kunert) + ZZ postprocessing → residual based, explicitanisotropic error estimator for the H1 norm.

Effectivity index aspect ratio independent whenever theestimator is equidistributed in the direction of maximum andminimum stretching.

Applied to a wide range of elliptic and parabolic problems,very well suited for problems with boundary or internal layers.


Perspectives

Goal oriented estimations → Formaggia, Micheletti, Perottoet al.

Polynomial Preserving Recovery (Naga Zhang 2004) insteadof ZZ post-processing.

Residual based, implicit error estimators?

Numerical linear algebra with anisotropic meshes?

Hyperbolic problems? Compressible Navier-Stokes?

Interpolation error induced by remeshing?

Convergence of anisotropic adaptive finite element algorithms?


adaptive finite elements with large aspect ratio: theory ... · adaptive nite elements with large...

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