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Page 1: Domain decomposition strategies with black box subdomain solvers

Domain decomposition strategies with black boxsubdomain solvers

Silvia Bertoluzza

Istituto di Matematica applicata e Tecnologie Informatiche del CNR, Pavia

Grenoble, May 19th, 2011

Silvia Bertoluzza (IMATI-CNR) DD with Black Box SD solvers Grenoble, May 19th, 2011 1 / 18

Page 2: Domain decomposition strategies with black box subdomain solvers

Motivations

Domain decomposition

Solution of differential problem on Ω = ∪Ωk reduced to solvingproblems in the Ωk ’s + coupling condition

Many advantages:

PreconditioningParallelizationPossibility of treating bigger problem: divide et impera

Two classes

Conforming: continuity is imposed strongly across the subdomain edgesNon conforming: continuity is imposed weakly across the subdomainedges

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Page 3: Domain decomposition strategies with black box subdomain solvers

Motivations

Non conforming methods

Allows to couple discretizations of different type:

FE + spectral methodswavelets + FEFE + FE on non matching grids

Flexibility in the choice of meshes / Adaptivity

Possibility of using the “locally best” method

Allows to use different models in different subdomain

Different approaches: Mortar method, 3 Fields formulation, Lagrangemultiplier approach, . . .

Drawback: coupling condition −→ saddle point problem: stabilityproblems and limitations due to inf-sup conditions

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Page 4: Domain decomposition strategies with black box subdomain solvers

Motivations

Bulk of the computation: solution of local problems

Great effort spent on optimization of mono-domain codes for solvingPDE’s

Two level of optimization

Algorithm optimization (minimize computational cost)Architecture dependent optimization at compiling time

Goal

Develop domain decomposition methods allowing the use of optimizedcodes as subdomain solvers in a (non conforming) domain decompositionmethod

Important: black-box SD solver

Do not want to touch the SD code in any way

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Page 5: Domain decomposition strategies with black box subdomain solvers

Motivations

Use black box SD solvers

Many problems to be faced

Informatics problems: need to be able two make two differentinstances of a code (or two different codes!) talk to each other. Hugeproblem

Mathematical problems

design global solution strategy using standard local solversgive conditions ensuring convergence of the iterative strategydesign suitable preconditionersneed to be able to verify that the SD solvers verify the conditions forconvergence

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Page 6: Domain decomposition strategies with black box subdomain solvers

Motivations

To fix the ideas

Model problem−∆u = f in Ω

u = 0 on ∂Ω

Domain decomposition

−∆u = f in Ωk , ∀k[u] = 0 on Σ

[∂νu] = 0 on Σ

+ BC ([·] =jump)

Geometry

Ω = ∪kΩk

Γk = ∂Ωk \ ∂Ωk

Σ = ∪Γk

ΩΩ

Ω

Γ Γ3

Ω

Γ1

1

3

2 2Ω

Ω

Γ Γ3

Σ

Ω

Γ1

1

3

2 2

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Page 7: Domain decomposition strategies with black box subdomain solvers

Motivations

The Steklov-Poincare approach

Observation: if we knew u on Σ, Dirichlet problems on thesubdomains would give us u on all the Ωk ’s!

φh: guess for u|Σ → −∆ Solver → uk(φh) (guess for u on Ωk)

Is φh a good guess? Check r = [∂νu(φh)]. If r ∼ 0 then guess isgood!

Steklov-Poincare based methods: find φh “minimizing” |r |

Basis for many DD methods

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Page 8: Domain decomposition strategies with black box subdomain solvers

The FETI approach

What can codes do?

Steklov-Poincare operator: “guess” u|Σ −→ “residual”: [∂νu|Σ]

Problem: computation of ∂ν of solution not always available

Input: what are we free to prescribe

differential operatorr.h.s. f(non homogeneous) Dirichlet boundary conditions(non homogeneous) Neumann boundary conditions

Output: what can we expect to retrievevalue of the solution at (boundary) nodesouter normal derivative: more rare!

To be as general as possible design a domain decomposition strategyusing Neumann solvers and imposing continuity of the solution.

Silvia Bertoluzza (IMATI-CNR) DD with Black Box SD solvers Grenoble, May 19th, 2011 8 / 18

Page 9: Domain decomposition strategies with black box subdomain solvers

The FETI approach

Idea: switch the role of trace and normal derivative

“guess” λ = ∂νu|Σsolve Neumann problems on subdomains∫

Ωk

∇u · ∇v =

∫Ωk

f v +

∫Γk

(χkλ)v

(χk = ±1 depends on whether ν is the outer or inner normal w.r. to Ωk)

−→ “residual”: r = [u]|Σ with u solution of Neumann problems

minimize |r | by some iterative technique (Ex: CG method)

floating domains → solution determined modulo a constant

use FETI approach −→ saddle point problem with a scalar multiplier perfloating domain

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Page 10: Domain decomposition strategies with black box subdomain solvers

The FETI approach

Discretization

Galerkin discretization with λ ∈ Λh ⊂ L2(Σ) finite dimensional

Discrete problem is well posed provided Λh ⊇ Λ0h

Λ0h = λ ∈ L2(Σ) : ∀`, k λ|Γk∩Γ` = constant

→ use any suitable method

Projected Conjugate Gradient methodConjugate Gradient method on a reduced space

Replace solution of Neumann problem with black box call to a PDEnumerical solver A+

δ

Skip analysis

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Page 11: Domain decomposition strategies with black box subdomain solvers

The FETI approach

Assumptions

The discrete solver A+δ satisfy the following estimate:

‖(A+δ − A+)f ‖1,∗ . δs‖f ‖s−1

(weaker assumption) The discrete solver A+h satisfy the following

estimate:‖B(A+

h − A+)f ‖1/2,∗ . δs‖f ‖s−1

The space Λh verify the following inverse inequality

‖λh‖s,Σ . h−s‖λh‖−1/2,Σ

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Page 12: Domain decomposition strategies with black box subdomain solvers

The FETI approach

Theorem

Assumptions

The discrete solver A+δ satisfy the following estimate:

‖(A+δ − A+)f ‖1,∗ . δs‖f ‖s−1

The space Λh verify the following inverse inequality

‖λh‖s,Σ . h−s‖λh‖−1/2,Σ

the meshsizes for the subdomain problems are sufficiently finer thanthe meshsize of Λh.

Then the matrix S corresponding to the discrete problem is symmetricpositive definite, provided. Moreover we have that

‖λ− λh‖−1/2,Σ . (hs + δs)‖u‖s+1

S.B. 2011Silvia Bertoluzza (IMATI-CNR) DD with Black Box SD solvers Grenoble, May 19th, 2011 12 / 18

Page 13: Domain decomposition strategies with black box subdomain solvers

The FETI approach

Numerical experiments: 2D problem

Discretization for Λ: piecewise constants on a uniform mesh

Sobdomain solvers: Matlab PDE toolbox (functions used: meshinit,assempde)

Test: true solution u = (x + 1)2 + y sin(x + 1) on Ω = (−1, 2)× (0, 1)

Boundary conditions: homogeneous Dirichlet on x = −1,non-homogeneous Neuman elsewhere

-1 0 2

ν

-1 -0.5 0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

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Page 14: Domain decomposition strategies with black box subdomain solvers

The FETI approach

h = 1/N meshsize for Λh

δ = 1/M1 meshsize on Ω1

δ = 1/M2 meshsize on Ω2

5 situations

(a) M1 = M2 = N − 1,(b) M1 = M2 = N,(c) M1 = M2 = N + 1,(d) M1 = N + 1, M2 = N − 1(e) M1 = 2N, M2 = [1.5N]

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Page 15: Domain decomposition strategies with black box subdomain solvers

The FETI approach

N Case (a) Case (b) Case (c) Case (d) Case (e)

5 - .10093 .08280 .10539 .06271110 - .05002 .050023 .05043 .0288520 - .02416 .02416 .02459 .0143740 - .01201 .01201 .01198 .00718

Table: Errors for u

N Case (a) Case (b) Case (c) Case (d) Case (e)

5 - .04082 .03013 .05866 .0203410 - .01747 .01081 .03873 .0075920 - .01756 .00839 .04806 .0033440 - .00470 .00350 .07849 .00145

Table: Errors for λ

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Page 16: Domain decomposition strategies with black box subdomain solvers

The FETI approach

Numerical experiment 3D

Discretization for Λ: cubic splines

Sobdomain solvers: Comsol+Matlab (functions used: )

Right hand side f = x + 3 ∗ y on Ω = (−1, 2)× (0, 1)× (0, 1)

Boundary conditions: homogeneous Dirichlet on x = −1,homogeneous Neumann elsewhere

Comsol: Domain and Grid Comsol: Soluzione

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Page 17: Domain decomposition strategies with black box subdomain solvers

The FETI approach

Convergence of the CG algorithm

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Page 18: Domain decomposition strategies with black box subdomain solvers

The FETI approach

Conclusions

Different strategies available to use standard solvers in domaindecomposition

Possibility of using Dirichlet or Neumann solver

Inf-sup or inf-sup type conditions need to be satisfied

Stabilization techniques that do not touch the solvers are available

Open issues

Preconditioning (easy)Monitoring the validity of the assumptionsExtending to other class of problems

Thank you!!

Silvia Bertoluzza (IMATI-CNR) DD with Black Box SD solvers Grenoble, May 19th, 2011 18 / 18