@let@token CALCULS RECENTS EN NORM-ORIENTED€¦ · CALCULS RECENTS EN NORM-ORIENTED G. Br ethes,...

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CALCULS RECENTS EN NORM-ORIENTED G. Br` ethes * , O. Allain ** , Alain Dervieux * (*)INRIA - projet Ecuador Sophia-Antipolis, France (**) Lemma Engineering, France [email protected] April 7, 2015

Transcript of @let@token CALCULS RECENTS EN NORM-ORIENTED€¦ · CALCULS RECENTS EN NORM-ORIENTED G. Br ethes,...

Page 1: @let@token CALCULS RECENTS EN NORM-ORIENTED€¦ · CALCULS RECENTS EN NORM-ORIENTED G. Br ethes, O. Allain , Alain Dervieux (*)INRIA - projet Ecuador Sophia-Antipolis, France (**)

CALCULS RECENTS ENNORM-ORIENTED

G. Brethes∗, O. Allain∗∗, Alain Dervieux∗

(*)INRIA - projet Ecuador Sophia-Antipolis, France(**) Lemma Engineering, France

[email protected]

April 7, 2015

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Motivations

−div(1

ρ∇u) = f

Norm-oriented adaptation

Adaptative FMG

Smooth test case: boundary layer

Singular test case: discontinuous coefficient

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Norm-Oriented mesh adaptation

Minimize:

j(Ωh) = ||u − uh||2L2(Ωh)

Term u − uh is approximated by a corrector u′.

A priori Corrector:

a(u′prio , φh) =∑∂Tij

[∇φh]jumpTi ,Tj

· nij

∫∂Tij

(Πhu − u) dσ

+(f , φh)− (fh, φh)

with u approximated by a quadratic function.u′prio approximates Πhu − uh:

u′prio = u′prio − (πhuh − uh).

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Norm-Oriented mesh adaptation

Novel corrector : Defect Correction:Ωh,Ωh/2:

uh = A−1h fh , uh/2 = A−1

h/2fh/2 ⇒ u − uh/2 ≈1

4(u − uh)

Finer-grid Defect-Correction (DC) corrector:

Ahu′DC =

4

3Rh/2→h(Ah/2Ph→h/2uh − fh/2)

u′DC approximates Πhu − uh:

u′DC = u′DC − (πhuh − uh).

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Norm-Oriented Adaptation

We introduce an adjoint:

ψ ∈ Ω, a(ψ, u∗) = (u′, ψ)

a(u′, u∗) = (u′, u′)

a(u − uh, u∗) = j(Ωh)

Mopt,norm = K1(|ρ(H(u∗))|, uM)

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Norm-Oriented Adaptation

Step 1:

a(u′prio , φh) =∑∂Tij

[∇φh]jumpTi ,Tj

· nij

∫∂Tij

(Πhu − uh) dσ

+(f , φM)− (fh, φh)

or

Ahu′DC =

4

3Rh/2→h(Ah/2Ph→h/2uh − fh/2)

u′ = u′ − (πhuh − uh).

Step 2:

a(ψ, u∗) = (u′, ψ)

Step 3:Mopt,norm = K1(|ρ(H(u∗))|, uM)

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Adaptative FMG

The mesh adaptation loop isapplied as anintermediate loopbetween:

- the FMG phases loop and- the MG cycling.

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Adaptative FMG

The Hessian case relies only on the solution of the discret PDE.We apply a stopping MG criterion using mesh convergence andresidual.The stopping criterion used for the Hessian case cannot directly beapplied to the case of norm-oriented since the two other states,- corrector and- adjointtend to zero when N increases.New stopping criteria are being tested.

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Smooth test case: boundary layer

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Smooth test case: boundary layer

Fully 2D Boundary layer test case : comparison of error cuts fory = 0.5: plus signs (+) depict the approximation error u − uh andcrosses (×) depict the a priori corrector u′prio . The corrector is ableto correct about 60% of the approximation error.

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Smooth test case: boundary layer

Fully 2D Boundary layer test case : comparison of error cuts fory = 0.5: plus signs (+) depict the approximation error u − uh andcrosses (×) depict the Defect-Correction corrector u′DC . Thecorrector is able to correct about 95% of the approximation error.

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Smooth test case: boundary layer

Fully 2D Boundary layer test case: convergence of the error norm|u − uh|L2 as a function of number of vertices in the mesh for (+)non-adaptative FMG, (×) Hessian-based adaptative FMG, (∗)norm-oriented adaptative FMG.

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Smooth test case: boundary layer

Fully 2D Boundary layer test case: convergence based on (×) the apriori corrector or on the (∗) Defect-Correction one, compared with() a virtual adaptation controlled by u − uh.

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Smooth test case: boundary layer

Fully 2D Boundary layer test case: comparison of the predictederror norm proposed by the method with deffect correction, curvewith (), with the exact error norm (×).

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Singular test case: discontinuous coefficient

Poisson problem with discontinuous coefficient: sketch of exactsolution definition and a typical computation of it.

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Singular test case: discontinuous coefficient

Singular test case : comparison of error cuts for y = 0.5: plus signs(+) depict the approximation error u − uh and crosses (×) depictthe a priori corrector u′prio .

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Singular test case: discontinuous coefficient

Singular test case : comparison of error cuts for y = 0.5: plus signs(+) depict the approximation error u − uh and crosses (×) depictthe Defect-Correction corrector u′DC .

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Singular test case: discontinuous coefficient

Poisson problem with discontinuous coefficient: convergence of theerror norm |u − uh|L2 as a function of number of vertices in themesh for (+) non-adaptative FMG, (×) Hessian-based adaptativeFMG and (∗) norm-oriented adaptative FMG.

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Singular test case: discontinuous coefficient

Poisson problem with discontinuous coefficient: efficiency analysis :CPU time for the norm-oriented adaptative FMG based on (×) thea priori corrector or on the (∗) Defect-Correction one, comparedwith () a virtual adaptation controlled by u − uh.

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Conclusions

The norm-oriented is improving:- new corrector, more accurate,- first test for FMG,- new applications: the discontinuous case produces second-orderconvergence are predicted by theory.

Open problems remain:- solve the FMG efficiency issue,- validate a good measure of the approximation error.

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