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Adaptive finite elements with large aspect ratio :theory and practice
M. Picasso
Institut d’Analyse et Calcul ScientifiqueEcole Polytechnique Federale de Lausanne
Switzerland
Crete 2008
M. Picasso Adaptive finite elements with large aspect ratio
Example 1 : 2D advection-diffusion
Mesh Isovalues
Boundary layer 10−4, 241 vertices, asp. ratio O(105), sameprecision with isotropic mesh O(105) vertices, SISC 2003.
Design of the stabilization parameter : anisotropic a priorierror estimates, with S. Micheletti and S. Perotto, SINUM2003.
M. Picasso Adaptive finite elements with large aspect ratio
Example 2 : 3D dendritic growth
Parabolic nonlinear system, existence E. Burman and J.Rappaz, M2AS 2003.Anisotropic a posteriori error estimates, with E. Burman,Interfaces Free Bound. 2003.Dendritic growth with convection, with J. Narski, CMAME2007, Fluid Dyn. Mater. Proc. 2007.Animation MeshAdapt mesh generator, INRIA Gammaproject, Distene company.
M. Picasso Adaptive finite elements with large aspect ratio
Example 3 : 3D supersonic flow around an aircraft
With Y. Bourgault (Ottawa), collaboration with INRIAGamma project, F. Alauzet, A. Loseille, supported by DassaultAviation, IJNMF 2008.
Compressible Euler solver (F. Alauzet), Anisotropic meshgenerator (C. Dobrzynski, P. Frey), Anisotropic errorestimator.
Animation, 106 vertices, single 32b processor.
M. Picasso Adaptive finite elements with large aspect ratio
Example 4 : Microfluidics
With V. Prachittham, unsteady convection-diffusion flows.
Space-time adaptive finite element with large aspect ratio.
Optimal a posteriori error estimates for the Crank Nicolsonscheme.
Akrivis Nochetto Makridakis, Math. Comp. 2006.
Animation.
Animation.
M. Picasso Adaptive finite elements with large aspect ratio
Outline
The anisotropic error estimator for the Laplace problem.
The anisotropic error estimator for the heat problem with theCrank Nicolson scheme.
M. Picasso Adaptive finite elements with large aspect ratio
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.
M. Picasso Adaptive finite elements with large aspect ratio
Anisotropic interpolation estimates
Formaggia Perotto, Numer. Math. 2001, 2003.
See also 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 =
(rT1,K
rT2,K
)=
(1 00 1
), ΛK =
(λ1,K 0
0 λ2,K
)=
(H 00 h
).
M. Picasso Adaptive finite elements with large aspect ratio
Anisotropic interpolation estimates
Formaggia Perotto, Numer. Math. 2001, 2003.
See also Kunert Verfurth, Numer. Math. 2000.
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
M. Picasso Adaptive finite elements with large aspect ratio
Anisotropic interpolation estimates
Formaggia Perotto, Numer. Math. 2001, 2003.
See also Kunert Verfurth, Numer. Math. 2000.
The number of neighbours must be bounded aboveindependently of h.
There is a (technical) restriction of the reference patchT−1
K (∆K ) which must be O(1). This excludes meshes having“large curvature”. In practice, anisotropic mesh generatorsexclude such meshes.
M. Picasso Adaptive finite elements with large aspect ratio
The anisotropic a posteriori error estimator
Upper bound : ∃C > 0 indep. mesh size and aspect ratio s.t.∫Ω
|∇(u − uh)|2dx
≤ C∑K∈Th
((∫K
f 2
)1/2
+1
2
(|∂K |
λ1,Kλ2,K
)1/2(∫∂K
[∇uh · n]2)1/2
)(
λ21,K
(rT1,KGK (u − uh)r1,K
)+ λ2
2,K
(rT2,KGK (u − uh)r2,K
))1/2
,
with GK (u − uh) = GK (e) =
∫
K
(∂e
∂x1
)2 ∫K
∂e
∂x1
∂e
∂x2∫K
∂e
∂x1
∂e
∂x2
∫K
(∂e
∂x2
)2
.
Lower bound if the mesh equidistributes the error in the directionsof maximum and minimum stretching :
λ21,K
(rT1,KGK (u − uh)r1,K
)≤ λ2
2,K
(rT2,KGK (u − uh)r2,K
)∀K ∈ Th.
M. Picasso Adaptive finite elements with large aspect ratio
The anisotropic a posteriori error estimator
The error estimator (e = u − uh) :
η2K =
(‖f ‖L2(K) +
1
2
(|∂K |
λ1,Kλ2,K
)1/2
‖ [∇uh · n] ‖L2(∂K)
)(
λ21,K
(rT1,KGK (e)r1,K
)+ λ2
2,K
(rT2,KGK (e)r2,K
))1/2
.
How to approach GK (e) =
∫
K
(∂e
∂x1
)2 ∫K
∂e
∂x1
∂e
∂x2∫K
∂e
∂x1
∂e
∂x2
∫K
(∂e
∂x2
)2
?
Zienkiewicz-Zhu (Πh is an approximate L2 projection onto Vh)∫K
(∂e
∂x1
)2
=
∫K
(∂(u − uh)
∂x1
)2
→∫
K
(∂uh
∂x1− Πh
∂uh
∂x1
)2
.
M. Picasso Adaptive finite elements with large aspect ratio
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
NKSufficient 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
M. Picasso Adaptive finite elements with large aspect ratio
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 (e)
2D : use the BL2D mesh generator (INRIA, Borouchaki, Laug)
3D : use the MeshAdapt mesh generator (INRIA, George,Distene) or the mmg3d mesh generator (Dobrzynski, Frey).
M. Picasso Adaptive finite elements with large aspect ratio
Adaptive meshes for the Laplace problem in 2D
Initial 10× 10 mesh
M. Picasso Adaptive finite elements with large aspect ratio
Adaptive meshes for the Laplace problem in 2D
TOL = 0.25 : adapted mesh after 30 mesh generations, 145 vertices
M. Picasso Adaptive finite elements with large aspect ratio
Effectivity index with anisotropic adapted meshes
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 2880.03125 10812 0.062 2.79 0.95 4250.015625 42562 0.031 2.79 0.98 1199
The effectivity index is aspect ratio independent on adaptedmeshes
M. Picasso Adaptive finite elements with large aspect ratio
The heat equation with Crank-Nicolson scheme
Optimal a posteriori error estimates for Crank-Nicolson timediscretization : Akrivis Makridadkis Nochetto, Math. Comp.2006.
With V. Prachittham and A. Lozinski :∂u
∂t−∆u = f , time
and space discretization.
For n = 1, ...,N, find unh ∈ Vh such that, for all vh ∈ Vh
1
τn
∫Ω(un
h−un−1h )vdx+
1
2
∫Ω∇(un−1
h +unh)·∇vdx =
1
2
∫Ω(f n−1+f n)vdx .
uhτ (x , t) =t − tn−1
τnunh(x) +
tn − t
τnun−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 .
M. Picasso Adaptive finite elements with large aspect ratio
Two possible time reconstructions
The Two-Point Time Reconstruction : following AkrivisMakridadkis Nochetto, Math. Comp. 2006
uhτ (x , t) = uhτ (x , t)+1
2(t−tn−1)(t−tn)wn
h , t ∈ In, n = 1 · · ·N,
where wnh ∈ Vh is defined by∫
Ωwn
h vh dx =
∫Ω
f n − f n−1
τn−∫
Ω∇∂uhτ
∂t· ∇vh dx , ∀vh ∈ Vh.
M. Picasso Adaptive finite elements with large aspect ratio
Two possible time reconstructions
The Three-Point Time Reconstruction : for tn−1 ≤ t ≤ tn
uhτ (x , t) = uhτ (x , t) +1
2(t − tn−1)(t − tn)∂2
nuh,
where
∂2nuh =
unh − un−1
h
τn−
un−1h − un−2
h
τn−1
(τn + τn−1)/2.
or equivalently
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 .
M. Picasso Adaptive finite elements with large aspect ratio
Error indicator
Approach
∫ T
t1
∫Ω
|∇(u − uhτ )|2 byN∑
n=2
∑K∈Th
η2K ,n with η2
K ,n =
∫ tn
tn−1
(‖f − ∂nuh + ∆uhτ‖L2(K) +
1
2λ1/22,K
‖[∇uhτ · n]‖L2(∂K)
)(
λ21,K
(rT1,KGK (u − uhτ )r1,K
)+ λ2
2,K
(rT2,KGK (u − uhτ )r2,K
))1/2
+
∥∥∥∥f − (f n−1/2 + (t − tn−1/2)f n − f n−2
τn + τn−1
)∥∥∥∥2
L2(K)
+τ 4n
∥∥∇∂2nuh
∥∥2
L2(K)
dt.
M. Picasso Adaptive finite elements with large aspect ratio
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 Mes. AR0.125 0.03 0.99 2.87 2.68 155 142 84 630.0625 0.015 0.99 2.89 2.91 348 201 52 1080.03125 0.0078 0.99 2.96 2.99 892 285 52 1650.015625 0.0040 1.00 2.88 2.71 4408 401 40 118
Conclusion : Vert = O(TOL−2) and N = O(TOL−1/2).
M. Picasso Adaptive finite elements with large aspect ratio
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