Domain decomposition strategies with black box subdomain solvers
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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 (IMATICNR) 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 (IMATICNR) 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 infsup conditions
Silvia Bertoluzza (IMATICNR) 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 monodomain 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: blackbox SD solver
Do not want to touch the SD code in any way
Silvia Bertoluzza (IMATICNR) 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 (IMATICNR) 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 (IMATICNR) DD with Black Box SD solvers Grenoble, May 19th, 2011 6 / 18

Motivations
The SteklovPoincare 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!
SteklovPoincare based methods: find h minimizing r 
Basis for many DD methods
Silvia Bertoluzza (IMATICNR) DD with Black Box SD solvers Grenoble, May 19th, 2011 7 / 18

The FETI approach
What can codes do?
SteklovPoincare 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 (IMATICNR) 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 = usolve 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 (IMATICNR) 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 (IMATICNR) 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 (IMATICNR) 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 (IMATICNR) 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,nonhomogeneous 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 (IMATICNR) 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 (IMATICNR) 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 (IMATICNR) 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 (IMATICNR) DD with Black Box SD solvers Grenoble, May 19th, 2011 16 / 18

The FETI approach
Convergence of the CG algorithm
Silvia Bertoluzza (IMATICNR) 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
Infsup or infsup 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 (IMATICNR) DD with Black Box SD solvers Grenoble, May 19th, 2011 18 / 18
MotivationsThe FETI approach