Some examples in domain decomposition

Transcription

1 Some examples in domain decomposition Frédéric Nataf nataf Laboratoire J.L. Lions, CNRS UMR7598. Université Pierre et Marie Curie, France Joint work with V. Dolean (Nice University) F. Hecht (Labo JLL). 1

2 Outline of the talk 1. Schwarz algorithm 2. Freefem++ implementation 3. Other algorithms 4. Conclusion 2

3 Schwarz method Two subdomain case The original Schwarz Method (H.A. Schwarz, 1870) (u) = f in Ω u = 0 on Ω. Schwarz Method : (u n 1,u n 2) (u n+1 1,u n+1 2 ) with (u n+1 1 ) = f in Ω 1 u n+1 1 = 0 on Ω 1 Ω u n+1 1 = u n 2 on Ω 1 Ω 2. (u n+1 2 ) = f in Ω 2 u n+1 2 = 0 on Ω 2 Ω u n+1 2 = u n+1 1 on Ω 2 Ω 1. Parallel algorithm, converges Preconditioner for Krylov type methods 3

4 Discrete setting Let T h be a mesh of a domain Ω and V h be a finite element space on it. For simplicity, we consider a Poisson problem: Find u h V h such that for any v V h u h v h = Ω Ω f v h. This yields a linear algebra problem Find U R dim(v h) such that: AU = F. We solve this equation by a Schwarz type domain decomposition method defined as most as possible at the discrete level. 4

5 Discrete setting Let (T h,i ) 1 i N define an overlapping mesh decomposition of the initial mesh T h and Ω i := K Th,i K. Define V h,i the finite element space on T h,i and the restriction operator r i : r i : V h V h,i u h u h Ω i Let R i be the boolean matrix corresponding to the restriction operator r i : R i := R i : R dim(vh) R dim(v h,i) 5

6 Discrete setting We have R i : R dim(v h) R dim(v h,i) and the transpose is a prolongation operator Ri T : R dim(vh,i) R dim(vh). The local Dirichlet matrices are given by A i := R i ARi T. We also need a kind of partition of unity defined by matrices D i D i : R dim(vh,i) R dim(v h,i) so that we have: N Ri T D i R i = Id i=1 6

7 Discrete setting The classical Schwarz preconditioners to be used in a Krylov method are ASM (additive Schwarz method) M 1 ASM := N i=1 RAS (restricted additive Schwarz) M 1 RAS := N i=1 R T i A 1 i R i R T i D i A 1 i R i The most costly part of the preconditioner is the solving of the local subproblems which can be done in parallel. The current FreeFem++ implementation is only sequential. Memory savings only. 7

8 FreeFem++ coding schwarz-laplace-coarse.edp Splits the mesh in non-overlapping subdomains either by a geometric manual partition Ph part; // piecewise constant function Ph xx=x,yy=y; part = int(xx/allongement*nn)*mm + int(yy*mm); // manual partition into nn*mm subdomains // cell i belongs to subdomain part[][i] or by a call to METIS interfaced via FreeFem++ load "metis"; int[int] nupart(th.nt); metisdual(nupart,th,npart); // partition into npart subdomains for(int i=0;i<nupart.n;++i) part[][i]=nupart[i]; // cell i belongs to subdomain part[][i] 8

9 FreeFem++ coding creating overlapping subdomains for(int i=0;i<npart;++i){ // non-overlapping partition Ph suppi= abs(part-i)<0.1; // boolean function 1 in the subdomain 0 elswhere Thin=aThn[i]=trunc(Th,suppi>0,label=10,split=1); // non-overlapping mesh, interfaces have label // overlapping partition suppi= abs(part-i)<0.1; AddLayers(Th,suppi[],sizeoverlaps,unssd[]); Thi=aTh[i]=trunc(Th,suppi>0,label=10,split=1); // overlapping mesh, interfaces have label 10 Rih[i]=interpolate(Vhi,Vh,inside=1); // Restriction operator : Vh -> Vhi 9

10 FreeFem++ coding RAS preconditioner M 1 RAS := N i=1 R T i D i A 1 i R i // RAS preconditionner func real[int] Mm1(real[int] &l){ // sum(rit *Ai^-1*Ri)*l s = 0; for(int i=0;i<npart;++i){ real[int] bi = Rih[i]*l; real[int] ui = A[i] ^-1 * bi; bi = Dih[i]*ui; s[] += Rih[i] *bi; } return s[]; } 10

11 FreeFem++ coding Other algorithms implementation of various classical methods RAS + Coarse grid corrections Schwarz methods with Robin interface conditions Neumann-Neumann or FETI preconditioners Neumann-Neumann or FETI preconditioners + Coarse grid corrections The Neumann-Neumann or FETI implementations in that no Lagrange multipliers nor pseudo-inverse were used. Can be used as a pedagogic or research tool (Joint work with V. Dolean and H. Xiang) 11

12 Joint work with V. Dolean and H. Xiang Design a Schwarz domain decomposition method for the Darcy equation div(κ u) = f that is robust with respect to the permeability field κ and a many domains decomposition: either manually or via Metis. viscosity IsoValue viscosity IsoValue Decomposition into 64 domains using Metis Figure 1: Permeability for multilayer and skyscraper cases Metis How to handle along the interface discontinuities in a simple and parallel manner? 12

13 Numerical results viscosity IsoValue Decomposition into 64 domains using Metis Figure 2: Permeability for the skyscraper case Metis decomposition uniform N N decomposition N N decomp. using Metis N 2 RAS RAS+Z Nico RAS+Z D2N RAS RAS+Z Nico RAS+Z D2N > >400 > > >400 >

14 Conclusion Key functions for the implementation of domain decomposition methods in Freefem++ are: metisdual trunc AddLayers interpolate It enabled to code in a sequential manner the Schwarz, Neumann-Neumann or FETI methods with coarse grid corrections. Pedagogic and research usefulness Saves in memory Next step is to have a parallel implementation 14