[PPT] - Convergence of the Adaptive Finite Element Method Carsten PowerPoint Presentation

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Convergence of the Adaptive Finite Element Method

Carsten Carstensen

Department of Mathematics, Humboldt-Universit¨ at zu Berlin http://www.math.hu-berlin.de/˜cc/ MATHEON, DFG Research Center

Chemnitz FEM Symposium

C. Carstensen (Humboldt)

Convergence of AFEM Chemnitz FEM 1 / 32

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Outline

Introduction: UFEM&AFEM, References AFEM: Algorithm for Energy Minimization Short History of Arguments to Prove Error Reduction Convergence Theory for Convex Minimisation Applications and Examples Thanks to sponsors DFG, FWF, EPSRC and to collaborators S. Bartels, R. Hoppe, A. Orlando, J. Valdman

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Introduction

UFEM

1 Universal algorithm (uniform for all data: RHSs etc.) 2 Tℓ+1 := red(Tℓ) 3 Convergence from limℓ→∞ hℓL∞(Ω) = 0, but convergence can be

arbitrarily bad (There is always some disaster RHS) AFEM

1 Specialised ”adapted” feedback algorithm (for ONE set of data) 2 Tℓ+1 generated from Tℓ, uℓ, all data, plus extra computations 3 Convergence is open since hℓL∞(Ω) → 0 is not guaranteed a priori!

Aims at error reduction or some other kind of convergence control

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Books on a posteriori FE error control

Eriksson-Estep-Hansbo-Johnson (1995) Verf¨ urth (1996) Ainsworth-Oden (2000) Babuska-Strouboulis (2001) Bangerth-Rannacher (2003) Neittaanm¨ aki-Repin (2004) ... and inside no theorem on AFEM convergence! (Except early results by Babuska et al. in 1D.)

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Papers on Convergence of AFEM in 2D

W. D¨
rfler: A convergent adaptive algorithm for Poisson’s equation.

SIAM Journal on Numerical Analysis 33 (1996) 1106–1124.

P. Morin, RH. Nochetto, and KG. Siebert: Local problems on stars:

a posteriori error estimation, convergence, and performance. Mathematics of Computation 72 (2003) 1067–1097 AND Convergence of adaptive finite element methods. SIAM Review 44 (2003) 631–658.

A. Veeser (2002): Convergent adaptive finite elements for the

nonlinear Laplacian. Numer. Math., 92, 4, 743–770.

P. Binev, W. Dahmen, and R. DeVore: Adaptive Finite Element

methods with Convergence Rates. Num. Math., 97(2) 219–268, (2004).

R. Stevenson: Optimality of AFEM, preprint 2005.

C, C-Hoppe, Braess-C-Hoppe

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(AFEM)

Input: coarse mesh T0 For ℓ = 0, 1, 2, . . . SOLVE ESTIMATE MARK REFINE Output: Sequence of nested discrete spaces V0 ⊆ V1 ⊆ V2 ⊆ . . . ⊆

∞

ℓ=0

Vℓ ⊆ V = W 1,p (Ω; Rm) with associated stress approximations (σℓ)ℓ∈N0. Task: Design an (AFEM) with limℓ→∞ σ − σℓLq = 0.

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SOLVE

Given shape regular triangulation Tℓ, define Vℓ := P1(Tℓ; Rm) ∩ V , compute some minimizer with Newton-Raphson scheme for E(vℓ) :=

Ω

W (Dvℓ) dx −

Ω

f · vℓ dx for all vℓ ∈ Vℓ. Compute discrete stress σℓ := DW (Duℓ) ∈ P0(Tℓ; Rm×n). W : Rm×n → R with pth order growth causes V = W 1,p (Ω; Rm), i.e., gradients in Lp and stresses in dual Lp′, 1/p + 1/p′ = 1. Remark: For class of degenerate convex minimization problems with p, p′, q, r, s, t, u and uℓ are non-unique, σ and σℓ are unique s.t. σ − σℓLr/t min

vℓ∈Vℓ

u − vℓW 1,p.

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ESTIMATE

Given interior edge E = ∂T+ ∩ ∂T− ∈ Eℓ, compute [σℓ] := σℓ|T+ − σℓ|T−, η(ℓ)

E

:= diam(E)1/p′[σℓ] · νELp′(E) and set ηℓ :=

E∈Eℓ

η(ℓ)

E p′1/p′

. Theorem (C 2006). There holds

T− T+ νE E

σ − σℓr

Lr/t(Ω) ηℓ + oscℓ

for oscillation

scp′

ℓ :=

z∈Kℓ
sc(f , ωz)p′.

For each inner node z with nodal basis function ϕz ∈ Vℓ, the patch reads ωz := {x ∈ Ω : 0 < ϕz(x)} and osc(f , ωz) := diam(ωz)f − fωzLp′(ωz).

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MARK

Bulk criterion with greedy algorithm. Sort Eℓ = {E1, . . . , EN} in list (E1, . . . , EN) s.t. η(ℓ)

E1 ≤ η(ℓ) E2 ≤ . . . ≤ η(ℓ) EN

and set k maximal with Θ ηp′

ℓ ≤ η(ℓ)p′ Ek

+ . . . + η(ℓ)p′

EN .

Then, for fixed 0 < Θ ≤ 1, Mℓ := {Ek, Ek+1, . . . , EN} satisfies ηp′

ℓ

E∈Mℓ

η(ℓ)

E p′

. Monitor oscℓ+1 to achieve at least limℓ→∞ oscℓ = 0.

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REFINE

Use concept of reference edge (i.e. opposite side of newest local vertex) E(T) for T = conv{E, F, G} and E, F, G ∈ Eℓ. Closure algorithm: Given shape-regular triangulation T := Tℓ and subset M := Mℓ of edges Eℓ in Tℓ, repeat (a)-(b) until T = ∅: (a) Choose T ∈ T with

E(T) ∩ M = ∅

AND E(T) ∈ M

and stop if there is no such T.

(b) While

T = ∅

AND E(T) ∈ M

do

(M := M ∪ {E(T)}, T := T \ {T}, T := N(T)). Then M := closure(Tℓ, Mℓ). Theorem (Bolte-C 2005+). M is minimal with Mℓ ⊆ M and ∀E ∈ M ∀T ∈ Tℓ(E) E(T) ∈ M.

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Red-Green-Blue Refinement with Reference Edge

Reference edge E(T) is bottom line and all M(T) are bisected via red(T) green(T) blueleft blueright bisect5(T) bisec3(T) Inner Node Property achieved via bisect5(T) for at least on neighbouring triangle of each edge in Mℓ.

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Mesh-Refinement

Maintain shape regularity. On coarse K ∈ T0, Tℓ|K is affine picture of reference triangle with solely right isosceles triangles. Tℓ allows for ℓ-independent H1-stable L2-projections (C 2004). Binev-Dahmen-DeVore 2004, Bolte-C 2005+ show card(Tℓ \ T0)

ℓ

j=1

card(Mj).

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Error Reduction Without Inner Node Property?

A counterexample to error reduction for W = ψ(| · |) and nearest-vertex-bisection on regular polygon:

T0 T1

Theorem (Bartels-C 2006+). Suppose Ω = T0 = T1 is regular polygon, decomposed in regular triangulations T0 and T1. Then, σ0 = σ1.

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History of Arguments to Prove Error Reduction

Linear elliptic PDE with energy norm | · | and solution u ∈ W 1,2 (Ω; Rm) (and stress field σ) respective discrete solution uℓ (and σℓ): (a) Reliability of error estimator [Rodriguez 94, C-Verf¨ urth SINUM 99] | u − uℓ |2

E∈Eℓ

hE

E

|[Duℓ]|2 ds + osc(f ; Tℓ)2 (b) Bulk criterion [D¨

rfler SINUM 96]

| u − uℓ |2

E∈Mℓ

hE

E

|[Duℓ]|2 ds + osc(f ; Tℓ)2 (c1) Discrete local efficiency [D¨

rfler SINUM 96] for E ∈ Mℓ

hE

E

|[Duℓ]|2 ds | uℓ+1 − uℓ |(ωE)2 + h2

Ef 2 L2(ωE )

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Cont. Arguments to Prove Error Reduction

(c2) Refined discrete local efficiency [Nochetto et al., Veeser] hE

E

|[Duℓ]|2 ds | uℓ+1 − uℓ |(ωE)2 + h2

Ef − fE2 L2(ωE )

(d) Finite overlap in (b)&(c2) yields | u − uℓ |2 ≤ C1 | uℓ+1 − uℓ |2 + C2 osc(f ; Tℓ)2 (e) Galerkin orthogonality | uℓ+1 − uℓ |2 = | u − uℓ |2 − | u − uℓ+1 |2 (f) Finish by rearranging C1| u − uℓ+1 |2 ≤ (C1 − 1) | u − uℓ |2 + C2 osc(f ; Tℓ)2 and division by C1 | u − uℓ+1 |2 ≤ (1 − C −1

1 ) |

u − uℓ |2 + C −1

1 C2 osc(f ; Tℓ)2

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Substitute of Inner Node Property

BAD OK Suppose free node z in Tℓ with patch ωz and edges E(z) in Tℓ|ωz. Assume all edges in E(z) are bisected but NOT all triangles in Tℓ(ωz) with bisec3 and reference edge on ∂ωz. Then

E∈E(z)

η2

E |

uℓ+1 − uℓ |2(Ωz) +

y ∈ ωz∩K
sc(f , ωy)2
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Adaptive Nonstandard FEM

Difficulty: Galerkin orthogonality fails for MFEM, but generalised Galerkin

rthogonality holds for fluxes p and pℓ

pℓ+1 − pℓ2 p − pℓ2 − p − pℓ+12 + osc(f ; Tℓ)2 and eventually allows for error reduction property [C-Hoppe (2006)] p − pℓ+12 ≤ ̺ p − pℓ2 + C osc(f ; Tℓ)2. Extra difficulty with nonconforming FEM: Balance 2 discrete equations! C-Hoppe: Convergence analysis of an adaptive edge finite element method for the 2D eddy current equations, 2005, J. Num. Math. C-Hoppe: Error reduction and convergence for an adaptive mixed finite element method, 2005, to appear in Math. Comp. C-Hoppe: Convergence analysis of an adaptive nonconforming finite element method, 2006, Numer. Math. 103: 251-266

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Convergence Rates by Rob Stevenson

1 Given 0 < λ < 1 and δ := λ|

u − uℓ | there exists mesh T δ

0 with FE

solution uδ

0 and |

u − uδ | ≤ δ plus card(T δ

0 ) ≤ δ−1/s |u|1/s As

2 Consider mesh T δ

ℓ := T δ 0 ∪ Tℓ coarsest refinement of T δ 0 and Tℓ with

set of edges Eδ

ℓ with FE solution uδ ℓ

3 Lemma A: card(Eℓ \ Eδ

ℓ ) card(T δ 0 )

4 Lemma B: |

uδ

ℓ − uℓ

| ηℓ(Eℓ \ Eδ

ℓ ) + osc(f , Tℓ)

5 Lemma C: For µℓ := osc(f , Tℓ)/|

u − uℓ | bdd and Θ sufficiently small, Eℓ \ Eδ

ℓ satisfies bulk criterion with Θ, whence

card(Mℓ) ≤ card(Eℓ \ Eδ

ℓ )

6 Refinement with card(TL) L

ℓ=0 card(Mℓ)

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Convergence Theory for Convex Minimisation

Given strongly convex W : Rm×n → R with quadratic growth and DW Lipschitz and RHS f ∈ L2(Ω; Rm), minimise E(v) :=

Ω

W (Dv(x)) dx −

Ω

f · v dx for v ∈ V := W 1,2 (Ω; Rm). Theorem (C 2006+). (a)-(c) are equivalent: (a) energy-reduction property δℓ+1 ≤ ρδℓ + h.o.t. for some ρ < 1; (b) discrete residual control κRℓV ∗ ≤ RℓV ∗

ℓ+1 + h.o.t.

for some κ > 0; (c) reliability of an hierarchical error estimator RℓV ∗ ≤ CηH + h.o.t. for some C > 0.

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Notation

There exist unique exact u ∈ V resp. uℓ ∈ Vℓ discrete minimizer with error u − uℓ2

V ≈ δℓ := E(uℓ) − E(u)

and Rℓ := −DE(uℓ). Let (ϕm : m ∈ M) be basis of complement Wℓ+1 of Vℓ in Vℓ+1 = Vℓ ⊕ Wℓ+1, Wℓ+1 := span(ϕm : m ∈ M). Define hierarchical estimator ηH = (

m∈M

η2

m)1/2

with ηm = Rℓ(ϕm)/ϕmV for each m ∈ M. M are marked edges and ϕm is (modified) nodal basis function of new node mid(E).

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Multiyield Plasticity

Generalisation of standard infinite elastoplasticity with different hardening mechanisms from C-Orlando-Valdmann (2005)

σ ε σ ε σ ε

σr

p

σr

b ,pr

Hr σ σ σ,e C τ σr

p

NEr(σr

p )

NEr(σr

p )={0}

Er Er σr

p

pr∈NEr(σr

p )

Rheological model for multisurface plasticity with hardening. Normal cones in evolution law on set of admissible stresses. AFEM with linear convergence of energy and stress errors

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Multiyield Plasticity

5 10 15 20 25 30 35 40 45 50 10 20 30 40 50 60

ΓN ΓN ΓD

uD=0 g=(0,gy) g=(0,0)

ΓN

g=(0,0) x y

(0,0) (0,44) (48,60) (48,44)

5 10 15 20 25 30 35 40 45 50 10 20 30 40 50 60

Cooks membrane with T0. T9 with 3 different phases.

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Convergence History for Multiyield Plasticity

10

1

10

2

10

3

10

4

10

5

10

6

10

−2

10

−1

10 Degrees of freedom, N η 1 0.5 1 0.3 Uniform refinement AFEM based on bulk criterion (Alg7.1) AFEM based on max refinement rule (Alg7.2) 5 10 15 20 25 30 35 40 10

−2

10

−1

10 Number of refinements loops η AFEM based on bulk criterion (Alg7.1) AFEM based on max refinement rule (Alg7.2)

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Convergence for (degenerate) Convex Minimization

Given W : Rm×n → R and f ∈ Lp(Ω; Rm), minimize E(v) :=

Ω

W (Dv(x)) dx −

Ω

f · v dx for v ∈ V := W 1,p (Ω; Rm). Class of degenerate convex energy densities with p, r, s, t for growth conditions |F|p − 1 W (F) |F|p + 1 for all F ∈ Rm×n and for convexity control (1 + |A|s + |B|s)−1|DW (A) − DW (B)|r W (B) − W (A) − DW (A; B − A) for all A, B ∈ Rm×n. Theorem (C-2006):

σ − σℓLr/t
ℓ=0,1,2,... ∈ ℓr
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Rexaled 2-Well Benchmark Example

With lower convex envelope to model macroscopic well-posed variables (e.g. stress and deformation) W (F) := max{0, |F|2 − 1}2 + 4(|F|2 − [F2 · F]2) C-Plechac (1997), C-Jochimsen (2003)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.5 1 1.5 −0.2 0.2 0.4 0.6 0.8 1 1.2

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Rexaled 2-Well Benchmark Example

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.5 1 1.5 −0.2 0.2 0.4 0.6 0.8 1

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AFEM with Reliability-Efficiency-Gap

10 10

1

10

2

10

3

10

4

10

5

10

−3

10

−2

10

−1

10 10

1

N 1 0.9 ηR (unif.) ηZ (unif.) ||σ −σh||4/3 (unif.) ηR (ηR−adapt.) ηZ (ηR−adapt.) ||σ −σh||4/3 (ηR−adapt.)

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Optimal Design Task

Topology optimisation leads to min

v∈H1

0(Ω)

Ω

ψ

|Dv|
dx −
Ω

v dx for µ1 = 1, µ2 = 2, 2t1 = t2, and ψ′(t) :=    µ2t for 0 ≤ t ≤ t1, t1µ2 = t2µ1 for t1 ≤ t ≤ t2, µ1t for t2 ≤ t.

λ = 0.0084 Square λ = 0.0145 L-Shape λ = 0.0284 Stop Sign λ = 0.0163 Slit

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Results from Bartels-C 2006

No oscillations δℓ := E(uℓ) - E(u) satisfies κ2−4 σ − σℓ4

L2 + δℓ+1 ≤ (1 − κδℓ)δℓ

for ℓ = 0, 1, 2, . . . and fixed 0 < κ ≤ 1. R-linear convergence in pre-asymptotic rate

σ − σℓL2
ℓ=0,1,2,... ∈ ℓ4 and (δℓ)ℓ=0,1,2,... ∈ ℓ2
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−1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1

Adaptively generated T0, T2, . . . , T10 of Ω4 = (−1, 1)2 \ [0, 1) × {0}.

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10

1

10

2

10

3

10

4

10

5

10

−5

10

−4

10

−3

10

−2

10

−1

N δ δl (uniform refinement) δl (adaptive refinement)

Decay of energy difference δℓ on the sequence of adaptively generated triangulations T0, T1, . . . T17 of Ω4 = (−1, 1)2 \ [0, 1) × {0}.

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Papers at website http://www.math.hu-berlin.de/˜cc/

C. Carstensen, A. Orlando, J. Valdman: A Convergent Adaptive Finite

Element Method for the Primal Problem of Elastoplasticity (2006) Journal for Numerical Methods in Engineering.

C. Carstensen, R. Hoppe: Error Reduction and Convergence for an

Adaptive Mixed Finite Element Method (2005) Math Comp

C. Carstensen, R. Hoppe: Convergence Analysis of an Adaptive Edge Finite

Element Method for the 2D Eddy Current Equations, 2005, J Num Math

C. Carstensen, R. Hoppe: Convergence Analysis of an Adaptive

Nonconforming Finite Element Method, 2006, Numer Math 103: 251-266.

S. Bartels, C. Carstensen: A Convergent Adaptive Finite Element Method

for an Optimal Design Problem. Preprint (2006).

D. Braess, C. Carstensen, R. Hoppe: Convergence Analysis of a

Conforming Adaptive Finite Element Method for an Obstacle Problem. Preprint (2006).

C. Carstensen: Convergence of Adaptive FEM for a Class of Degenerate

Convex Minimization Problems Preprint (2006).

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