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

Motivations

Domain decomposition

Solution of differential problem on = k reduced to solvingproblems in the k 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

Silvia Bertoluzza (IMATI-CNR) DD with Black Box SD solvers Grenoble, May 19th, 2011 2 / 18

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

Silvia Bertoluzza (IMATI-CNR) DD with Black Box SD solvers Grenoble, May 19th, 2011 3 / 18

Motivations

Bulk of the computation: solution of local problems

Great effort spent on optimization of mono-domain codes for solvingPDEs

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

Silvia Bertoluzza (IMATI-CNR) DD with Black Box SD solvers Grenoble, May 19th, 2011 4 / 18

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

Silvia Bertoluzza (IMATI-CNR) DD with Black Box SD solvers Grenoble, May 19th, 2011 5 / 18

Motivations

To fix the ideas

Model problem{u = f in u = 0 on

Domain decomposition

u = f in k , k[u] = 0 on

[u] = 0 on

+ BC ([] =jump)

Geometry

= kkk = k \ k = k

3

1

1

3

2 2

3

1

1

3

2 2

Silvia Bertoluzza (IMATI-CNR) DD with Black Box SD solvers Grenoble, May 19th, 2011 6 / 18

Motivations

The Steklov-Poincare approach

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

h: guess for u| Solver uk(h) (guess for u on k)

Is h a good guess? Check r = [u(h)]. If r 0 then guess isgood!

Steklov-Poincare based methods: find h minimizing |r |

Basis for many DD methods

Silvia Bertoluzza (IMATI-CNR) DD with Black Box SD solvers Grenoble, May 19th, 2011 7 / 18

The FETI approach

What can codes do?

Steklov-Poincare operator: guess u| residual: [u|]Problem: computation of 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

The FETI approach

Idea: switch the role of trace and normal derivative

guess = u|solve Neumann problems on subdomains

k

u v =

k

f v +

k

(k)v

(k = 1 depends on whether is the outer or inner normal w.r. to k) residual: r = [u]| with u solution of Neumann problems

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

floating domains solution determined modulo a constantuse FETI approach saddle point problem with a scalar multiplier perfloating domain

Silvia Bertoluzza (IMATI-CNR) DD with Black Box SD solvers Grenoble, May 19th, 2011 9 / 18

The FETI approach

Discretization

Galerkin discretization with h L2() finite dimensional

Discrete problem is well posed provided h 0h0h = { L2() : `, k |k` = constant}

use any suitable methodProjected Conjugate Gradient methodConjugate Gradient method on a reduced space

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

Skip analysis

Silvia Bertoluzza (IMATI-CNR) DD with Black Box SD solvers Grenoble, May 19th, 2011 10 / 18

The FETI approach

Assumptions

The discrete solver A+ satisfy the following estimate:

(A+ A+)f 1, . sf s1

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

B(A+h A+)f 1/2, . sf s1

The space h verify the following inverse inequality

hs, . hsh1/2,

Silvia Bertoluzza (IMATI-CNR) DD with Black Box SD solvers Grenoble, May 19th, 2011 11 / 18

The FETI approach

Theorem

Assumptions

The discrete solver A+ satisfy the following estimate:

(A+ A+)f 1, . sf s1The space h verify the following inverse inequality

hs, . hsh1/2,the meshsizes for the subdomain problems are sufficiently finer thanthe meshsize of h.

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

h1/2, . (hs + s)us+1S.B. 2011

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

The FETI approach

Numerical experiments: 2D problem

Discretization for : 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 = (1, 2) (0, 1)Boundary conditions: homogeneous Dirichlet on x = 1,non-homogeneous Neuman elsewhere

-1 0 2

1

-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

Silvia Bertoluzza (IMATI-CNR) DD with Black Box SD solvers Grenoble, May 19th, 2011 13 / 18

The FETI approach

h = 1/N meshsize for h

= 1/M1 meshsize on 1

= 1/M2 meshsize on 2

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]

Silvia Bertoluzza (IMATI-CNR) DD with Black Box SD solvers Grenoble, May 19th, 2011 14 / 18

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

Silvia Bertoluzza (IMATI-CNR) DD with Black Box SD solvers Grenoble, May 19th, 2011 15 / 18

The FETI approach

Numerical experiment 3D

Discretization for : cubic splines

Sobdomain solvers: Comsol+Matlab (functions used: )

Right hand side f = x + 3 y on = (1, 2) (0, 1) (0, 1)Boundary conditions: homogeneous Dirichlet on x = 1,homogeneous Neumann elsewhere

Comsol: Domain and Grid Comsol: Soluzione

Silvia Bertoluzza (IMATI-CNR) DD with Black Box SD solvers Grenoble, May 19th, 2011 16 / 18

The FETI approach

Convergence of the CG algorithm

Silvia Bertoluzza (IMATI-CNR) DD with Black Box SD solvers Grenoble, May 19th, 2011 17 / 18

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

MotivationsThe FETI approach