An introduction to Schwarz methods

Transcription

1 An introduction to Schwarz methods Victorita Dolean Université de Nice and University of Geneva École thematique CNRS Décomposition de domaine November 15-16, 2011

2 Outline 1 Introduction 2 Schwarz algorithms as solvers 3 Schwarz algorithms as preconditioners 4 Classical coarse grid method 5 Coarse grid for heterogeneous problems 6 An abstract 2-level Schwarz - the GenEO algorithm V. Dolean Schwarz methods 2 / 107

3 Motivation: pro and cons of direct solvers Complexity of the Gauss factorization Gauss d = 1 d = 2 d = 3 dense matrix O(n 3 ) O(n 3 ) O(n 3 ) using band structure O(n) O(n 2 ) O(n 7/3 ) using sparsity O(n) O(n 3/2 ) O(n 2 ) Different sparse direct solvers PARDISO ( SUPERLU ( xiaoye/superlu) SPOOLES ( MUMPS ( UMFPACK (http: // V. Dolean Schwarz methods 3 / 107

4 Why iterative solvers? Limitations of direct solvers In practice all direct solvers work well until a certain barrier: two-dimensional problems (100K unknowns) three-dimensional problems (10K unknowns). Beyond, the factorization cannot be stored in memory any more. To summarize: below a certain size, direct solvers are chosen. beyond the critical size, iterative solvers are needed. V. Dolean Schwarz methods 4 / 107

5 Why domain decomposition? Natural iterative/direct trade-off Parallel processing is the only way to have faster codes, new generation processors are parallel: dual, quadri core. Large scale computations need for an artificial decomposition Memory requirements, direct solvers are too costly Iterative solvers are not robust enough. New iterative/direct solvers are welcome : these are domain decomposition methods In some situations, the decomposition is natural Moving domains (rotor and stator in an electric motor) Strongly heterogeneous media Different physics in different subdomains V. Dolean Schwarz methods 5 / 107

6 Linear Algebra from the End User point of view Direct DDM Iterative Cons: Memory Pro: Flexible Pros: Memory Difficult to Naurally Easy to Pros: Robustness Cons: Robustness solve(mat,rhs,sol) Few black box routines solve(mat,rhs,sol) Few implementations of efficient DDM Multigrid methods: very efficient but may lack robustness, not always applicable (Helmholtz type problems, complex systems) and difficult to parallelize. V. Dolean Schwarz methods 6 / 107

7 Outline 1 Introduction 2 Schwarz algorithms as solvers Three continuous Schwarz algorithms Connection with the Block-Jacobi algorithm Discrete setting Iterative Schwarz methods Convergence analysis Schwarz methods using Freefem++ Schwarz algorithms as solvers 3 Schwarz algorithms as preconditioners 4 Classical coarse grid method 5 Coarse grid for heterogeneous problems V. Dolean Schwarz methods 7 / 107

8 The First Domain Decomposition Method The original Schwarz Method (H.A. Schwarz, 1870) (u) = f in Ω u = 0 on Ω. Ω 1 Ω 2 Schwarz Method : (u1 n, un 2 ) (un+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 = u2 n 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 but very slowly, overlapping subdomains only. The parallel version is called Jacobi Schwarz method (JSM). V. Dolean Schwarz methods 8 / 107

9 Continuous ASM and RAS - I The algorithm acts on the local functions (u i ) i=1,2. To make things global, we need: extension operators, E i, s.t. for a function w i : Ω i R, E i (w i ) : Ω R is the extension of w i by zero outside Ω i. partition of unity functions χ i : Ω i R, χ i 0 and χ i (x) = 0 for x Ω i and s.t. w = 2 E i (χ i w Ωi ). i=1 Let u n be an approximation to the solution to the global Poisson problem and u n+1 is computed by solving first local subproblems and then gluing them together. V. Dolean Schwarz methods 9 / 107

10 Continuous ASM and RAS - II Local problems to solve (u n+1 i ) = f in Ω i u n+1 i = 0 on Ω i Ω u n+1 i = u n on Ω i Ω 3 i. Two ways to glue solutions Using the partition of unity functions Restricted Additive Schwarz (RAS) u n+1 := 2 i=1 E i (χ i u n+1 i ). Not based on the partition of unity Additive Schwarz (ASM) 2 u n+1 := E i (u n+1 i ). i=1 V. Dolean Schwarz methods 10 / 107

11 Block Jacobi methods - I Let us consider a linear system: AU = F with a matrix A of size m m, a right handside F R m and a solution U R m where m is an integer. Let D be the diagonal of A, the Jacobi algorithm reads: or equivalently, DU n+1 = DU n + (b AU n ), U n+1 = U n + D 1 (b AU n ) = U n + D 1 r n, where r n is the residual of the equation. V. Dolean Schwarz methods 11 / 107

12 Block Jacobi methods - II We now define a block Jacobi algorithm. The set of indices {1,..., m} is partitioned into two sets N 1 := {1,..., m s } and N 2 := {m s + 1,..., m}. Let U 1 := U N1, U 2 := U N2 and similarly F 1 := F N1, F 2 := F N2. The linear system has the following block form: ( ) ( ) ( ) A11 A 12 U1 F1 = A 21 A 22 U 2 F 2 where A ij := A Ni N j, 1 i, j 2. V. Dolean Schwarz methods 12 / 107

13 Block Jacobi methods - III The block-jacobi algorithm reads: ( ) ( ) ( ) A11 0 U n+1 1 F1 A 12 U2 n =. (1) 0 A 22 U n+1 F 2 2 A 21 U1 n Let U n = (U1 n, Un 2 )T, algorithm (1) becomes ( ) ( ) A A12 U n+1 = F U n. (2) 0 A 22 A 21 0 On the other hand, it can be rewritten equivalently ( ) ( ) ( ) U n+1 1 U n 1 ( U n+1 = 1 A11 0 r n U n A 22 r2 n where r n := F AU n, r n i := r n N i, i = 1, 2. ) (3) V. Dolean Schwarz methods 13 / 107

14 Block-Jacobi compact form In order to have a more compact form, let us introduce R 1 the restriction operator from N into N 1 R 2 the restriction operator from N into N 2. The transpose operator R T 1 is an extension operator from N 1 into N and the same holds for R T 2. Notice that A ii = R i AR T i. Block-Jacobi in compact form U n+1 = U n + (R T 1 (R 1AR T 1 ) 1 R 1 + R T 2 (R 2AR T 2 ) 1 R 2 )r n. (4) where r n := F AU n, r n i := r n N i, i = 1, 2. V. Dolean Schwarz methods 14 / 107

15 Schwarz algorithms as block Jacobi methods - I Let Ω := (0, 1) and consider the following BVP u = f in Ω u(0) = u(1) = 0. discretized by a three point finite difference scheme on the grid x j := j h, 1 j m where h := 1/(m + 1). Let u j u(x j ), f j := f (x j ), 1 j m and the discrete problem AU = F, U = (u j ) 1 j m, F = (f j ) 1 j m. where A j j := 2/h 2 and A j j+1 = A j+1 j := 1/h 2. Let domains Ω 1 := (0, (m s + 1) h) and Ω 2 := (m s h, 1) define an overlapping decomposition with a minimal overlap of width h. V. Dolean Schwarz methods 15 / 107

16 Schwarz algorithms as block Jacobi methods - II The discretization of the JSM for domain Ω 1 reads un+1 1,j 1 2un+1 1,j + u n+1 1,j+1 = f j, 1 j m s h 2 u n+1 1,0 = 0 u n+1 1,m s+1 = un 2,m s+1 Solving for U n+1 1 = (u n+1 1,j ) 1 j ms corresponds to solving a Dirichlet boundary value problem in subdomain Ω 1 with Dirichlet data taken from the other subdomain at the previous step. Then, U n+1 1 and U n+1 1 satisfy A 11 U n A 12 U2 n = F 1, A 22 U n A 21 U1 n = F 2.. V. Dolean Schwarz methods 16 / 107

17 Schwarz as block Jacobi methods - III The discrete counterpart of the extension operator E 1 (resp. E 2 ) is defined by E 1 (U 1 ) = (U 1, 0) T (resp. E 2 (U 2 ) = (0, U 2 ) T ). χ 1 χ 2 Ω x 1 ms x ms+1 Ω 2 then E 1 (U 1 ) + E 2 (U 2 ) = E 1 (χ 1 U 1 ) + E 2 (χ 2 U 2 ) = ( U1 When the overlap is minimal, the discrete counterparts of the three Schwarz methods are equivalent to the same block Jacobi algorithm. U 2 ). V. Dolean Schwarz methods 17 / 107

18 Continuous level Ω and an overlapping decomposition Ω = N i=1 Ω i. A function u : Ω R. Restriction of u : Ω R to Ω i, 1 i N. The extension E i of a function Ω i R to a function Ω R. Partition of unity functions χ i, 1 i N. Discrete level A set of d.o.f. N and a decomposition N = N i=1 N i. A vector U R #N. The restriction R i U where U R #N and R i is a rectangular #N i #N boolean matrix. Extension R T i. Diagonal matrices with positive entries, of size #N i #N i s. t. Id = N i=1 RT i D i R i. V. Dolean Schwarz methods 18 / 107

19 Restrictions operators Let T h be a mesh of a domain Ω and u h some discretization of a function u which is the solution of an elliptic Dirichlet BVP. This yields a linear algebra problem Find U R #N s.t. A U = F. Define the restriction operator r i = E T i : r i : u h u h Ωi Let R i be the boolean matrix corresponding to the restriction operator r i : R i := R i : R #N R #N i V. Dolean Schwarz methods 19 / 107

20 Partition of unity We have R i : R #N R #N i and the transpose is a prolongation operator Ri T : R #N i R #N. 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 #N i R #N i so that we have: N Ri T D i R i = Id i=1 V. Dolean Schwarz methods 20 / 107

21 Two subdomain case: 1d algebraic setting Let N := {1,..., 5} be partitioned into N 1 := {1, 2, 3} and N 2 := {4, 5} N 1 N 2 Then, matrices R 1, R 2, D 1 and D 2 are: R 1 = and R 2 = D 1 = and D 2 = ( ) ( ) V. Dolean Schwarz methods 21 / 107

22 Consider now the case overlapping case N δ=1 1 := {1, 2, 3, 4} and N δ=1 2 := {3, 4, 5} N1 δ=1 N2 δ=1 Then, matrices R 1, R 2, D 1, D 2 are: R 1 = and R 2 = D 1 = / /2 0 and D 2 = 0 1/ /2 V. Dolean Schwarz methods 22 / 107

23 Two subdomain case: 1d finite element decomposition Partition of the 1D mesh corresponds to an ovr. decomp. of N : N 1 := {1, 2, 3} and N 2 := {3, 4, 5} Ω 1 Ω 2 Then, matrices R 1, R 2, D 1, D 2 are: R 1 = and R 2 = /2 0 0 D 1 = and D 2 = / V. Dolean Schwarz methods 23 / 107

24 Consider now the situation of an overlapping partition. N δ=1 1 := {1, 2, 3, 4} and N δ=1 2 := {2, 3, 4, 5} Ω δ=1 1 Ω δ=1 2 Then, matrices R 1, R 2, D 1, D 2 are: R 1 = and R 2 = / D 1 = 0 1/ /2 0 and D 2 = 0 1/ / / V. Dolean Schwarz methods 24 / 107

25 Multi-D and many subdomains: General procedure The set of indices N can be partitioned by an automatic graph partitioner such as METIS or SCOTCH. From the input matrix A, a connectivity graph is created. Two indices i, j N are connected if the matrix coefficient A ij 0. Even if matrix A is not symmetric, the connectivity graph is symmetrized. Algorithms that find a good partitioning of highly unstructured graphs are used. This distribution must be done so that the number of elements assigned to each processor is roughly the same (balance the computations among the processors). The number of adjacent elements assigned to different processors is minimized (minimize the communication between different processors). V. Dolean Schwarz methods 25 / 107

26 Multi-D algebraic setting Let us consider a partition into N subsets N := N i=1 N i, N i N j = for i j. N 1 N δ=1 1 N 2 N 3 N δ=1 2 N δ=1 3 Extend each subset N i with its direct neighbors to form Ni δ=1. Let R i be the restriction matrix from set N to the subset Ni δ=1 and D i be a diagonal matrix of size #Ni δ=1 #Ni δ=1, 1 i N such that for M j := {1 i N j N δ=1 i }. and j N δ=1 i, we define (D i ) jj := 1/#M j. V. Dolean Schwarz methods 26 / 107

27 Multi-D algebraic finite element decomposition In a FE setting, the computational domain is the union of elements of the finite element mesh T h. It is possible to create overlapping subdomains resolved by the finite element meshes: Ω i = τ for 1 i N. (5) τ T i,h V. Dolean Schwarz methods 27 / 107

28 Let {φ k } k N be a basis of the finite element space. For 1 i N, we define N i := {k N : supp(φ k ) Ω i }. For all degree of freedom k N, let µ k := # {j : 1 j N and supp(φ k ) Ω j }. Let R i be the restriction matrix from set N to the subset N i and D i be a diagonal matrix of size #N i #N i, 1 i N. Then, for k N i, we define (D i ) kk := 1/µ k. V. Dolean Schwarz methods 28 / 107

29 Algebraic formulation - JSM Define local unknowns U i := R i U for i = 1,..., N and local right handside F i := R i F. R i A U = R i A R T i (R i U) + R i A (Id R T i R i )U = F i = R i A R T i U i + R i A (Id R T i R i ) N j=1 RT j D j R j U = R i A R T i U i + N j=1 R i A (Id R T i R i )R T j D j U j Notice that (Id R T i R i )R T i R i = 0 so we have R i A R T i U i + j i R i A (Id R T i R i )R T j D j U j = F i (6) V. Dolean Schwarz methods 29 / 107

30 Algebraic formulation - JSM Let us define the block matrix à (extended matrix) (Ã) ii := R i A R T i, (Ã) ij := R i A (Id R T i R i )R T j D j, 1 i j N Define (extended) unknown vector and right-hand side Ũ := (U 1,..., U i,..., U N ) T R N i=1 #N i, F := (R 1 F,..., R i F,..., R N F) T R N i=1 #N i. Let (M JSM ) ii := (Ã) ii = R i A R T i. The block Jacobi method applied to the (extended) system is the JSM: à Ũ = F Ũ n+1 = Ũn + M 1 JSM r n, r n := F à Ũn. (7) V. Dolean Schwarz methods 30 / 107

31 Algebraic formulation - RAS and ASM As for (RAS), we give the following definition M 1 N RAS := Ri T D i (R i A Ri T ) 1 R i (8) i=1 so that the iterative RAS algorithm reads: U n+1 = U n + M 1 RAS r n, r n := F A U n. For (ASM), we give the following definition M 1 N ASM := Ri T (R i A Ri T ) 1 R i (9) i=1 so that the iterative ASM algorithm reads: U n+1 = U n + M 1 RAS r n. V. Dolean Schwarz methods 31 / 107

32 Geometrical analysis in 1d Let L > 0 and the domain Ω = (0, L) be decomposed into two subodmains Ω 1 := (0, L 1 ) and Ω 2 := (l 2, L) with l 2 L 1. The error e n i := u n i u Ωi, i = 1, 2 satisfies d 2 e n+1 1 dx 2 = 0 in (0, L 1 ) e n+1 e n+1 1 (0) = 0 1 (L 1 ) = e n 2 (L 1) then, d 2 e n+1 2 dx 2 = 0 in (l 2, L) e n+1 2 (l 2 ) = e n+1 1 (l 2 ) e n+1 2 (L) = 0. (10) Thus the errors are affine functions in each subdomain: e n+1 1 (x) = e n 2 (L 1) x L 1 and e n+1 2 (x) = e n+1 1 (l 2 ) L x L l 2. V. Dolean Schwarz methods 32 / 107

33 Thus, we have e n+1 2 (L 1 ) = e n+1 1 (l 2 ) L L 1 L l 2 = e n 2 (L 1) l 2 L 1 L L 1 L l 2. Let δ := L 1 l 2 denote the size of the overlap, we have e n+1 2 (L 1 ) = l 2 l 2 + δ L l 2 δ L l 2 e n 2 (L 1) = 1 δ/(l l 2) 1 + δ/l 2 e n 2 (L 1). It is clear that δ > 0 is sufficient and necessary to have convergence. e 1 1 e 1 2 e 2 1 e 2 2 e 3 1 x 0 l 2 L 1 L V. Dolean Schwarz methods 33 / 107

34 Fourier analysis in 2d - I Let R 2 decomposed into two half-planes Ω 1 = (, δ) R and Ω 2 = (0, ) R with an overlap of size δ > 0 and the problem (η )(u) = f in R 2, u is bounded at infinity, By linearity, the errors e n i := u n i u Ωi satisfy the JSM f = 0: (η )(e n+1 1 ) = 0 in Ω 1 e n+1 1 is bounded at infinity (11) e n+1 1 (δ, y) = e2 n(δ, y), (η )(e n+1 2 ) = 0 in Ω 2 e n+1 2 is bounded at infinity (12) e n+1 2 (0, y) = e1 n(0, y). V. Dolean Schwarz methods 34 / 107

35 Fourier analysis in 2d - II By taking the partial Fourier transform of the equation in the y direction we get: ) (η 2 x 2 + k 2 (ê n+1 1 (x, k)) = 0 in Ω 1. For a given k, the solution ê n+1 1 (x, k) = γ+ n+1 (k) exp(λ+ (k)x) + γ n+1 (k) exp(λ (k)x). must be bounded at x =. This implies and similarly, ê n+1 1 (x, k) = γ n+1 + (k) exp(λ+ (k)x) ê n+1 2 (x, k) = γ n+1 (k) exp(λ (k)x) V. Dolean Schwarz methods 35 / 107

36 Fourier analysis in 2d - III From the interface conditions we get γ n+1 + (k) = γn (k) exp(λ (k)δ), γ n+1 (k) = γn +(k) exp( λ + (k)δ). Combining these two and denoting λ(k) = λ + (k) = λ (k), we get for i = 1, 2, γ n+1 ± (k) = ρ(k; α, δ)2 γ n 1 ± (k) with ρ the convergence rate given by: where λ(k) = η + k 2. ρ(k; α, δ) = exp( λ(k)δ), (13) V. Dolean Schwarz methods 36 / 107

37 Fourier analysis in 2d - IV ρ Remark We have the following properties: For all k R, ρ(k) < exp( η δ) < 1 so that γ n i (k) 0 uniformly as n goes to infinity. ρ 0 as k tends to infinity, high frequency modes of the error converge very fast. When there is no overlap (δ = 0), ρ = 1 and there is stagnation of the method. k V. Dolean Schwarz methods 37 / 107

38 About FreeFem++ (survival kit) FreeFem++ allows a very simple and natural way to solve a great variety of variational problems (FEM, DG). It is possible to have access to the underlying linear algebra such as the stiffness or mass matrices. Tutorial: php?article239. A very detailed documentation of FreeFem++ is available on the official website V. Dolean Schwarz methods 38 / 107

39 Let a homogeneous Dirichlet boundary value problem for a Laplacian defined on a unit square Ω =]0, 1[ 2 : { u = f dans Ω u = 0 sur Ω The variational formulation of the problem (14) Find u H 1 0 (Ω) := {w H1 (Ω) : w = 0, on Γ 1 Γ 2 Γ 3 Γ 4 } such that Ω u. vdx f v dx = 0, v H0 1 (Ω). Ω Feature of Freefem++: penalization of Dirichlet BC. Let TGV (Très Grande Valeur in French) be a very large value, the above variational formulation is approximated by Find u H 1 (Ω) such that u. vdx + TGV u v fv dx = 0, v H 1 (Ω). Ω i=1,...,4 Γ i Ω V. Dolean Schwarz methods 39 / 107

40 The following FreeFem++ script is solving this problem // Number of mesh points in x and y directions int Nbnoeuds=10; The text after // symbols are comments ignored by the FreeFem++ language. Each new variable must be declared with its type (here int designs integers). //Mesh definition mesh Th=square(Nbnoeuds,Nbnoeuds,[x,y]); The function square returns a structured mesh of the square, the sides of the square are labelled from 1 to 4 in trigonometrical sense. V. Dolean Schwarz methods 40 / 107

41 Define the function representing the right hand side // Function of x and y func f=x*y; and the P 1 finite element space Vh over the mesh Th. // Finite element space on the mesh Th fespace Vh(Th,P1); //uh and vh are of type Vh Vh uh,vh; The functions u h and v h belong to the P 1 finite element space V h which is an approximation to H 1 (Ω). // variational problem definition problem heat(uh,vh,solver=lu)= int2d(th)(dx(uh)*dx(vh)+dy(uh)*dy(vh)) -int2d(th)(f*vh) +on(1,2,3,4,uh=0); V. Dolean Schwarz methods 41 / 107

42 The keyword problem allows the definition of a variational problem (without solving it) u h. v h dx + TGV u h v h fv h dx = 0, v h V h. Ω i=1,...,4 Γ i Ω where TGV is equal to The parameter solver sets the method that will be used to solve the resulting linear system. To solve the problem we need //Solving the problem heat; // Plotting the result plot(uh,wait=1); The Freefem++ script can be saved with your favourite text editor (e.g. under the name heat.edp). In order to execute the script write the shell command FreeFem++ heat.edp The result will be displayed in a graphic window. V. Dolean Schwarz methods 42 / 107

43 Solve Neumann or Fourier boundary conditions such as u + u = f dans Ω u n = 0 sur Γ 1 (15) u = 0 sur Γ 2 u n + αu = g sur Γ 3 Γ 4 The new variational formulation consists in determining u h V h such that u h. v h dx + Ω αu h v h + TGV Γ 3 Γ 4 u h.v h Γ 2 gv h fv h dx = 0, v h V h. Γ 3 Γ 4 Ω The Freefem++ definition of the problem problem heat(uh,vh)= int2d(th)(dx(uh)*dx(vh)+dy(uh)*dy(vh)) +int1d(th,3,4)(alpha*uh*vh) -int1d(th,3,4)(g*vh) -int2d(th)(f*vh) +on(2,uh=0); V. Dolean Schwarz methods 43 / 107

44 In order to use some linear algebra package, we need the matrices. The keyword varf allows the definition of a variational formulation varf heat(uh,vh)= int2d(th)(dx(uh)*dx(vh)+dy(uh)*dy(vh)) +int1d(th,3,4)(alpha*uh*vh) -int1d(th,3,4)(g*vh) -int2d(th)(f*vh) +on(2,uh=0); matrix Aglobal; // stiffness sparse matrix Aglobal = heat(vh,vh,solver=umfpack);// UMFPACK solver Vh rhsglobal; //right hand side vector rhsglobal[] = heat(0,vh); Here rhsglobal is a FE function and the associated vector of d.o.f. is rhsglobal[]. The linear system is solved by using UMFPACK // Solving the problem by a sparse LU sover uh[] = Aglobalˆ-1*rhsglobal[]; V. Dolean Schwarz methods 44 / 107

45 Decomposition into overlapping domains I Suppose we want a decomposition of a rectangle Ω into nn mm domains with approximately nloc points in one direction. int nn=4,mm=4; int npart= nn*mm; int nloc = 20; real allong = 1; allong = real(nn)/real(mm); mesh Th=square(nn*nloc*allong,mm*nloc,[x*allong,y]); fespace Vh(Th,P1); fespace Ph(Th,P0); Ph part; Ph xx=x,yy=y; part = int(xx/allong*nn)*mm + int(yy*mm); plot(part,fill=1,value=1,wait=1,ps="decompunif.eps"); V. Dolean Schwarz methods 45 / 107

46 For arbitrary decompositions, use METIS or SCOTCH. int nn=4,mm=4; int npart= nn*mm; int nloc = 20; real allong = 1; allong = real(nn)/real(mm); mesh Th=square(nn*nloc*allong,mm*nloc,[x*allong,y]); fespace Vh(Th,P1); fespace Ph(Th,P0); Ph part; bool withmetis = 1; if(withmetis) // Metis partition { load "metis"; int[int] nupart(th.nt); metisdual(nupart,th,npart); for(int i=0;i<nupart.n;++i) part[][i]=nupart[i]; } plot(part,fill=1,value=1,wait=1,ps="decompmetis.eps"); V. Dolean Schwarz methods 46 / 107

47 Decomposition into overlapping domains II To build the overlapping decomposition and the associated algebraic call the routine SubdomainsPartitionUnity. Output: overlapping meshes ath[i] the restriction/interpolation operators Rih[i] from the local finite element space Vh[i] to the global one Vh the diagonal local matrices Dih[i] from the partition of unity. include "createpartition.edp"; include "decompmetis.edp"; // overlapping partition int sizeovr = 3; mesh[int] ath(npart); // sequence of ovr. meshes matrix[int] Rih(npart); // local restriction operators matrix[int] Dih(npart); // partition of unity operators SubdomainsPartitionUnity(Th,part[],sizeovr,aTh,Rih,Dih); V. Dolean Schwarz methods 47 / 107

48 RAS and ASM: global data We first need to define the global data. // Solve Dirichlet subproblem Delta (u) = f // u = 1 on the global boundary int[int] chlab=[1,1,2,1,3,1,4,1 ]; Th=change(Th,refe=chlab); macro Grad(u) [dx(u),dy(u)] func f = 1; func g = 1; // EOM // right hand side // Dirichlet data // global problem Vh rhsglobal,uglob; varf vaglobal(u,v) = int2d(th)(grad(u) *Grad(v)) +on(1,u=g) + int2d(th)(f*v); matrix Aglobal; Aglobal = vaglobal(vh,vh,solver = UMFPACK); rhsglobal[] = vaglobal(0,vh); uglob[] = Aglobalˆ-1*rhsglobal[]; // matrix // rhs V. Dolean Schwarz methods 48 / 107

49 RAS and ASM: local data And then the local problems // overlapping partition int sizeovr = 4; mesh[int] ath(npart); // overlapping meshes matrix[int] Rih(npart); // restriction operators matrix[int] Dih(npart); // partition of unity SubdomainsPartitionUnity(Th,part[],sizeovr,aTh,Rih,Dih); matrix[int] aa(npart); // Dirichlet matrices for(int i = 0;i<npart;++i) { cout << " Domain :" << i << "/" << npart << endl; matrix at = Aglobal*Rih[i] ; aa[i] = Rih[i]*aT; set(aa[i],solver = UMFPACK);// direct solvers } V. Dolean Schwarz methods 49 / 107

50 RAS and ASM : Schwarz iteration cout << "Iter: " << iter << " Err = " << err << endl plot(un,wait=1,value=1,fill=1,dim=3,ps = "solution.eps") int nitermax = 1000; Vh un = 0, rn = rhsglobal; // initial guess for(int iter = 0;iter<nitermax;++iter) {real err = 0; Vh er = 0; for(int i = 0;i<npart;++i) {real[int] bi = Rih[i]*rn[]; // restriction real[int] ui = aa[i] ˆ-1 * bi; // local solve bi = Dih[i]*ui; // bi = ui; ASM er[] += Rih[i] *bi; } un[] += er[]; // build new iterate rn[] = Aglobal*un[]; // global residual rn[] = rn[] - rhsglobal[]; rn[] *= -1; err = sqrt(er[] *er[]); if(err < 1e-5) break; plot(un,wait=1,value=1,fill=1,dim=3);} V. Dolean Schwarz methods 50 / 107

51 Convergence Convergence history of the RAS solver for different values of the overlapping parameter overlap= overlap=5 overlap= Note that this convergence, not very fast even in a simple configuration of 4 subdomains. The iterative version of ASM does not converge. For this reason, the ASM method is always used a preconditioner for a Krylov method such as CG, GMRES or BiCGSTAB. V. Dolean Schwarz methods 51 / 107

52 Outline 1 Introduction 2 Schwarz algorithms as solvers 3 Schwarz algorithms as preconditioners Neumann series and Krylov spaces Krylov methods Application to DDM Schwarz preconditioners using FreeFEM++ 4 Classical coarse grid method 5 Coarse grid for heterogeneous problems 6 An abstract 2-level Schwarz - the GenEO algorithm V. Dolean Schwarz methods 52 / 107

53 Fixed point method Consider a well-posed but difficult to solve linear system A x = b and B an easy to invert matrix of the same size than A. A possible iterative method is a fixed point algorithm x n+1 = x n + B 1 (b A x n ) and x is a fixed point of the operator: x x + B 1 (b A x). Let r 0 := b Ax 0 and C := B 1 A, a direct computation shows that we have: n x n = (I d C) i B 1 r 0 + x 0. (16) i=0 We have convergence iff the spectral radius of the matrix I d C is smaller than one. V. Dolean Schwarz methods 53 / 107

54 Why Krylov methods I Consider now a preconditioned Krylov applied to the linear system: B 1 A x = B 1 b Let us denote x 0 an initial guess and r 0 := B 1 b C x 0 the initial residual. Then y := x x 0 solves C y = r 0. The basis for Krylov methods is the following Lemma Let C be an invertible matrix of size N N. Then, there exists a polynomial P of degree p < N such that C 1 = P(C). V. Dolean Schwarz methods 54 / 107

55 Proof. Let be a minimal polynomial of C of degree d N: M(X) := d a i X i i=0 We have d i=0 a i C i = 0 and there is no non zero polynomial of lower degree that annihilates C. Thus, a 0 cannot be zero since C d a i C i 1 = 0 i=1 d a i C i 1 = 0. i=1 Then, d 1 i=0 a i+1 X i would be an annihiling polynomial of C of degree lower than d. This implies I d + C d i=1 a i a 0 C i 1 = 0 C 1 := d i=1 a i a 0 C i 1. V. Dolean Schwarz methods 55 / 107

56 Coming back to the linear system, we have x = x 0 + d i=1 ( a i a 0 )C i 1 r 0. Thus, it makes sense to introduce Krylov spaces, K n (C, r 0 ) K n (C, r 0 ) := Span{r 0, Cr 0,..., C n 1 r 0 }, n 1. to seek y n an approximation to y. Example: The CG methods applies to symmetric positive definite (SPD) matrices and minimizes the A 1 -norm of the residual when solving Ax = b: CG { Find y n K n (A, r 0 ) such that A y n r 0 A 1 = min w K n (A,r 0 ) A w r 0 A 1. V. Dolean Schwarz methods 56 / 107

57 A detailed analysis reveals that x n = y n + x 0 can be obtained by the quite cheap recursion formula: for i = 1, 2,... do ρ i 1 = (r i 1, r i 1 ) 2 if i = 1 then p 1 = r 0 else β i 1 = ρ i 1 /ρ i 2 p i = r i 1 + β i 1 p i 1 end if q i = Ap i 1 α i = ρ i 1 (p i, q i ) 2 x i = x i 1 + α i p i r i = r i 1 α i q i check convergence; continue if necessary end for V. Dolean Schwarz methods 57 / 107

58 Preconditioned Krylov By solving an optimization problem: GMRES { Find y n K n (C, r 0 ) such that C y n r 0 2 = min w K n (C,r 0 ) C w r 0 2 a preconditioned Krylov solve will generate an optimal x n K in K n (C, B 1 r 0 ) := x 0 + Span{B 1 r 0, C B 1 r 0,..., C n 1 B 1 r 0 }. This minimization problem is of size n. When n is small w.r.t. N, its solving has a marginal cost. Thus, xk n has a computing cost similar to that of x n. But, since x n K n (B 1 A, B 1 r 0 ) as well but with frozen coefficients, we have that x n is less optimal (actually much much less) than xk n. V. Dolean Schwarz methods 58 / 107

59 Schwarz methods as preconditioners In the previous Krylov methods we can use as preconditioner RAS (in conjunction with BiCGStab or GMRES) N B 1 := M 1 RAS = i=1 R T i D i (R i A R T i ) 1 R i ASM (in a CG methods) N B 1 := M 1 ASM = i=1 R T i (R i A R T i ) 1 R i V. Dolean Schwarz methods 59 / 107

60 Preconditioner in CG We use M 1 ASM as a preconditioner a Krylov method: conjugate gradient since M 1 symmetric. ASM and A are At iteration m the error for the PCG method is bounded by: x x m M 1 2 ASM AM 1 2 ASM where κ is the condition number of M 1 solution. [ ] m κ 1 2 x x 0 κ + 1 M ASM AM 2 ASM ASM A and x is the exact V. Dolean Schwarz methods 60 / 107

61 The CG with the ASM preconditioner becomes: for i = 1, 2,... do ρ i 1 = (r i 1, M 1 ASM r i 1) 2 if i = 1 then p 1 = M 1 ASM r 0 else β i 1 = ρ i 1 /ρ i 2 p i = M 1 ASM r i 1 + β i 1 p i 1 end if q i = Ap i 1 ρ i 1 α i = (p i, q i ) 2 x i = x i 1 + α i p i r i = r i 1 α i q i check convergence; continue if necessary end for V. Dolean Schwarz methods 61 / 107

62 The action of the global operator is given by Vh rn, s; func real[int] A(real[int] &l) { rn[]= Aglobal*l; return rn[]; } // A*u The preconditioning method can be Additive Schwarz (ASM) func real[int] Mm1(real[int] &l) { s = 0; for(int i=0;i<npart;++i) { mesh Thi = ath[i]; real[int] bi = Rih[i]*l; real[int] ui = aa[i] ˆ-1 * bi; } s[] += Rih[i] *ui; } return s[]; // restricts rhs // local solves // prolongation The Krylov method applied V. Dolean in thisschwarz casemethods is the Conjugate 62 / 107

63 The Krylov method applied in this case is the CG. The performance is now less sensitive to the overlap size overlap= overlap=5 overlap= V. Dolean Schwarz methods 63 / 107

64 We can also use RAS as a preconditioner, by taking into account the partition of unity func real[int] Mm1(real[int] &l) { s = 0; for(int i=0;i<npart;++i) { mesh Thi = ath[i]; real[int] bi = Rih[i]*l; } // restricts rhs real[int] ui = aa[i] ˆ-1 * bi; // local solves bi = Dih[i]*ui; s[] += Rih[i] *bi; // prolongation } return s[]; this time in conjuction with BiCGStab since we deal with non-symmetric problems. V. Dolean Schwarz methods 64 / 107

65 Weak scalability How to evaluate the efficiency of a domain decomposition? Weak scalability definition How the solution time varies with the number of processors for a fixed problem size per processor. It is not achieved with the one level method Number of subdomains ASM The iteration number increases linearly with the number of subdomains in one direction. V. Dolean Schwarz methods 65 / 107

66 Convergence curves- more subdomains Plateaus appear in the convergence of the Krylov methods Log_10 (Error) x4 8x M2 2x2 2x2 M2 4x4 M2 8x Number of iterations (GCR) V. Dolean Schwarz methods 66 / 107

67 How to achieve scalability Stagnation corresponds to a few very low eigenvalues in the spectrum of the preconditioned problem. They are due to the lack of a global exchange of information in the preconditioner. u = f in Ω u = 0 on Ω The mean value of the solution in domain i depends on the value of f on all subdomains. A classical remedy consists in the introduction of a coarse grid problem that couples all subdomains. This is closely related to deflation technique classical in linear algebra (see Nabben and Vuik s papers in SIAM J. Sci. Comp, 200X). V. Dolean Schwarz methods 67 / 107

68 Outline 1 Introduction 2 Schwarz algorithms as solvers 3 Schwarz algorithms as preconditioners 4 Classical coarse grid method Coarse grid correction Theoretical convergence result Deflation and coarse grid Classical coarse grid using FreeFEM++ 5 Coarse grid for heterogeneous problems 6 An abstract 2-level Schwarz - the GenEO algorithm V. Dolean Schwarz methods 68 / 107

69 Adding a coarse grid We add a coarse space correction (aka second level) Let V H be the coarse space and z be a basis, V H = span z, writing R 0 = Z T we define the two level preconditioner as: M 1 ASM,2 := RT 0 (R 0AR T 0 ) 1 R 0 + N i=1 R T i A 1 i R i. The Nicolaides approach is to use the kernel of the operator as a coarse space, this is the constant vectors, in local form this writes: Z := (Ri T D i R i 1) 1 i N where D i are chosen so that we have a partition of unity: N Ri T D i R i = Id. i=1 V. Dolean Schwarz methods 69 / 107

70 Theoretical convergence result Theorem (Widlund, Sarkis) Let M 1 ASM,2 be the two-level additive Schwarz method: κ(m 1 ASM,2 (1 A) C + H ) δ where δ is the size of the overlap between the subdomains and H the subdomain size. This does indeed work very well Number of subdomains ASM ASM + Nicolaides V. Dolean Schwarz methods 70 / 107

71 Idea of the proof (Upper bound) Lemma If each point in Ω belongs to at most k 0 of the subdomains Ω j, then the largest eigenvalue of M 1 ASM,2 A satisfies λ max (M 1 ASM,2 A) k Assumption (Stable decomposition) There exists a constant C 0, such that every u V admits a decomposition u = N i=0 RT i u i, u i V i, i = 0,..., N that satisfies: N ã i (u i, u i ) C0 2 a(u, u). i=0 V. Dolean Schwarz methods 71 / 107

72 Idea of the proof (Lower bound) Theorem If every v V admits a C 0 -stable decomposition (with uniform C 0 ), then the smallest eigenvalue of M 1 AS,2 A satisfies λ min (M 1 ASM,2 A) C 2 0. Therefore, the condition number of the two-level Schwarz preconditioner can be bounded by κ(m 1 ASM,2 A) C2 0 (k 0 + 1). V. Dolean Schwarz methods 72 / 107

73 Deflation and Coarse grid correction Let A be a SPD matrix, we want to solve Ax = b with a preconditioner M (for example the Schwarz method). Let Z be a rectangular matrix so that the bad eigenvectors belong to the space spanned by its columns. Define P := I AQ, Q := ZE 1 Z T, E := Z T AZ, Examples of coarse grid preconditioners P A DEF2 := P T M 1 +Q, P BNN := P T M 1 P+Q (Mandel, 1993) Some properties: QAZ = Z, P T Z = 0 and P T Q = 0. Let r n be the residual at step n of the algorithm: Z T r n = 0. How to choose Z? V. Dolean Schwarz methods 73 / 107

74 Coarse grid correction for smooth problems For a Poisson like problem, Nicolaides (1987), Sarkis (2002). Let (χ i ) 1 i N denote a partition of unity : 10 4 SCHWARZ additive Schwarz with coarse gird acceleration χ Z =. χ χ N X: 25 Y: 1.658e V. Dolean Schwarz methods 74 / 107

75 Coarse grid implementation - I It is enough to replace the Schwarz preconditioner by P BNN as follows. First build E = Z T AZ Vh[int] Z(npart); for(int i=0;i<npart;++i) { Z[i]=1.; real[int] zit = Rih[i]*Z[i][]; real[int] zitemp = Dih[i]*zit; Z[i][]=Rih[i] *zitemp; } real[int,int] Ef(npart,npart); for(int i=0;i<npart;++i) { real[int] vaux = A(Z[i][]); for(int j=0;j<npart;++j) Ef(j,i) = Z[j][] *vaux; } matrix E; E = Ef; set(e,solver=umfpack); // E = ZˆT*A*Z V. Dolean Schwarz methods 75 / 107

76 Coarse grid implementation - II Then the coarse space correction Q = ZE 1 Z T : func real[int] Q(real[int] &l) { real[int] res(l.n); res=0.; real[int] vaux(npart); for(int i=0;i<npart;++i) { vaux[i]=z[i][] *l; } real[int] zaux=eˆ-1*vaux; for(int i=0;i<npart;++i) { res +=zaux[i]*z[i][]; } return res; } // Q = Z*Eˆ-1*ZˆT // zaux=eˆ-1*zˆt*l // Z*zaux V. Dolean Schwarz methods 76 / 107

77 Coarse grid implementation - III The projector out of the coarse space P = I QA and its transpose P T : func real[int] P(real[int] &l) { real[int] res=q(l); real[int] res2=a(res); res2 -= l; res2 *= -1.; return res2; } func real[int] PT(real[int] &l) { real[int] res=a(l); real[int] res2=q(res); res2 -= l; res2 *= -1.; return res2; } // P = I - A*Q // PˆT = I-Q*A V. Dolean Schwarz methods 77 / 107

78 Coarse grid implementation - IV And finally the preconditioner P BNN = P T M 1 P + Q: int j; func real[int] BNN(real[int] &u) { real[int] aux1 = Q(u); real[int] aux2 = P(u); real[int] aux3 = Mm1(aux2); aux2 = PT(aux3); aux2 += aux1; ++j; return aux2; } // precond BNN V. Dolean Schwarz methods 78 / 107

79 Outline 1 Introduction 2 Schwarz algorithms as solvers 3 Schwarz algorithms as preconditioners 4 Classical coarse grid method 5 Coarse grid for heterogeneous problems The heterogeneous coefficient case Coarse grid for problems with high heterogeneities The DtN algorithm Theoretical and numerical results 6 An abstract 2-level Schwarz - the GenEO algorithm V. Dolean Schwarz methods 79 / 107

80 Motivation Large discretized system of PDEs strongly heterogeneous coefficients (high contrast, nonlinear, multiscale) E.g. Darcy pressure equation, P 1 -finite elements: Applications: flow in heterogeneous / stochastic / layered media structural mechanics electromagnetics etc. Au = f cond(a) α max α min h 2 Goal: iterative solvers robust in size and heterogeneities V. Dolean Schwarz methods 80 / 107

81 Darcy equation with heterogeneities (α(x, y) u) = 0 in Ω R 2, u = 0 on Ω D, u n = 0 on Ω N. IsoValue Decomposition α(x, y) Jump ASM ASM + Nicolaides V. Dolean Schwarz methods 81 / 107

82 Objectives Strategy Define an appropriate coarse space V H 2 = span(z 2 ) and use the framework previously introduced, writing R 0 = Z2 T the two level preconditioner is: P 1 ASM 2 := RT 0 (R 0AR T 0 ) 1 R 0 + N i=1 R T i A 1 i R i. The coarse grid must be Local (calculated on each subdomain) parallel Adaptive (calculated automatically) Easy and cheap to compute (on the boundary for instance) Robust (must lead to an algorithm whose convergence does not depend on the partition or the jumps in coefficients) V. Dolean Schwarz methods 82 / 107

83 Heuristic approach: what functions should be in Z 2? The error satisfies the Schwarz algorithm, it is harmonic, so it satisfies a maximum principle. Ω i 1 e n i Ω i Ω i+1 e n+1 i 1 e n+1 i+1 Fast convergence Ω i 1 e n i Ω i Ω i+1 e n+1 i 1 e n+1 i+1 Slow convergence Idea Ensure that the error decreases quickly on the subdomain boundaries which translates to making e Γi n i big. V. Dolean Schwarz methods 83 / 107

84 Ensuring that the error decreases quickly on the subdomain boundaries The Dirichlet to Neumann operator is defined as follows: Let g : Γ i R, DtN Ωi (g) = α v n, i Γi where v satisfies { ( div(α ))v = 0, in Ω i, v = g, on Ω i. To construct the coarse space, we use the low frequency modes associated with the DtN operator: DtN Ωi (vi λ ) = λ α vi λ with λ small. The functions vi λ are extended harmonically to the subdomains. V. Dolean Schwarz methods 84 / 107

85 Theoretical convergence result Suppose we have (v λ k i, λ k i ) 1 k n Γi the eigenpairs of the local DtN maps (λ 1 i λ 2 i...) and that we have selected m i in each subdomain. Then let Z be the coarse space built via the local DtN maps: Z := (R T i D i Ṽ λk i i ) 1 i N; 1 k mi Theorem (D., Nataf, Scheichl and Spillane 2010) Under the monotonicity of α in the overlapping regions: κ(m 1 ASM,2 A) C(1 + max 1 i N 1 δ i λ m i +1 i where δ i is the size of the overlap of domain Ω i and C is independent of the jumps of α. If m i is chosen so that, λ m i +1 i 1/H i the convergence rate will be analogous to the constant coefficient case. V. Dolean Schwarz methods 85 / 107 )

86 Results with the new DtN method Jump ASM ASM + Nicolaides ASM + DtN IsoValue Decomposition α(x, y) With DtN the jumps do not affect convergence We put at most two modes per subdomain in the coarse grid (using the automatic selection process) V. Dolean Schwarz methods 86 / 107

87 Numerical results IsoValue e e e e e e+06 IsoValue Channels and inclusions: 1 α , the solution and partitionings (Metis or not) V. Dolean Schwarz methods 87 / 107

88 Numerical results AS P BNN : AS + Z Nico P BNN : AS + Z D2N Error Iteration count ASM convergence for channels and inclusions 4 4 Metis partitioning V. Dolean Schwarz methods 88 / 107

89 Numerical results subdomain i m i total number of eigenvalues Metis 4 by 4 decomposition V. Dolean Schwarz methods 89 / 107

90 Numerical results ASM ASM+Nico ASM+DtN Metis Metis Metis Convergence results for the hard test case V. Dolean Schwarz methods 90 / 107

91 Numerical results Optimality #Z per subd. ASM ASM+Z Nico ASM+Z D2N max(m i 1, 1) 273 m i m i m i is given automatically by the strategy. Taking one fewer eigenvalue has a huge influence on the iteration count Taking one more has only a small influence V. Dolean Schwarz methods 91 / 107

92 Results for elasticity (Problem) E IsoValue sigma IsoValue Young s modulus (1 E 10 6 ) Poisson s ratio (0.35 ν 0.48) V. Dolean Schwarz methods 92 / 107

93 Results for 2d elasticity (Solution) 10 1 AS 10 0 P BNN : AS + Z Nico P BNN : AS + Z D2N GMRES P BNN : AS + Z D2N Error Iteration count Overlap is two grid cells V. Dolean Schwarz methods 93 / 107

94 Outline 1 Introduction 2 Schwarz algorithms as solvers 3 Schwarz algorithms as preconditioners 4 Classical coarse grid method 5 Coarse grid for heterogeneous problems 6 An abstract 2-level Schwarz - the GenEO algorithm Schwarz abstract setting Numerical results V. Dolean Schwarz methods 94 / 107

95 Problem setting I Given f (V h ) find u V h a(u, v) = f, v A u = f v V h Assumption throughout: A symmetric positive definite (SPD) Examples: Darcy Elasticity Eddy current a(u, v) = Ω κ u v dx a(u, v) = Ω C ε(u) : ε(v) dx a(u, v) = Ω ν curl u curl v + σ u v dx Heterogeneities / high contrast / nonlinearities in parameters V. Dolean Schwarz methods 95 / 107

96 Problem setting II 1 V h... FE space of functions in Ω based on mesh T h = {τ} 2 A given as set of element stiffness matrices + connectivity (list of DOF per element) Assembling property: a(v, w) = τ a τ (v τ, w τ ) where a τ (, ) symm. pos. semi-definite 3 {φ k } n k=1 (FE) basis of V h on each element: unisolvence set of non-vanishing basis functions linearly independent fulfilled by standard FE continuous, Nédélec, Raviart-Thomas of low/high order 4 Two more assumptions on a(, ) later! V. Dolean Schwarz methods 96 / 107

97 Schwarz setting I Overlapping partition: Ω = N j=1 Ω j (Ω j union of elements) V j := span { φ k : supp(φ k ) Ω j } such that every φ k contained in one of those spaces, i.e. N V h = j=1 Example: adding layers to non-overlapping partition (partition and adding layers based on matrix information only!) V j V. Dolean Schwarz methods 97 / 107

98 Schwarz setting II Local subspaces: V j V h j = 1,..., N Coarse space (defined later): V 0 V h Additive Schwarz preconditioner: M 1 ASM,2 = N j=0 R j A 1 j R j where A j = R j AR j and R j R j : V j V h natural embedding V. Dolean Schwarz methods 98 / 107

99 Partition of unity Definitions: dof (Ω j ) := { k : supp(φ k ) Ω j } idof (Ω j ) := { } k : supp(φ k ) Ω j imult(k) := # { j : k idof (Ω j ) } V j = span{φ k } k idof (Ωj ) Partition of unity: (used for design of coarse space and for stable splitting) Ξ j v = k idof (Ω j ) 1 imult(k) v k φ k for v = n v k φ k k=1 Properties: N Ξ j v = v j=1 Ξ j v V j V. Dolean Schwarz methods 99 / 107

100 Overlapping zone / Choice of coarse space Overlapping zone: Ω j = {x Ω j : i j : x Ω i } Observation: Ξ j Ωj \Ω j = id Coarse space should be local: V 0 = N j=1 V 0, j where V 0, j V j E.g. V 0, j = span{ξ j p j,k } m j k=1 V. Dolean Schwarz methods 100 / 107

101 Abstract eigenvalue problem Gen.EVP per subdomain: Find p j,k V h Ωj and λ j,k 0: a Ωj (p j,k, v) = λ j,k a Ω j (Ξ j p j,k, Ξ j v) v V h Ωj A j p j,k = λ j,k X j A j X j p j,k (X j... diagonal) (properties of eigenfunctions discussed soon) a D... restriction of a to D In the two-level ASM: Choose first m j eigenvectors per subdomain: V 0 = span { Ξ j p j,k } j=1,...,n k=1,...,m j V. Dolean Schwarz methods 101 / 107

102 Theory Two technical assumptions. Theorem (D., Hauret, Nataf, Pechstein, Scheichl, Spillane) If for all j: 0 < λ j,mj+1 < : [ κ(m 1 ASM,2 A) (1 + k 0) 2 + k 0 (2k 0 + 1) N max j=1 ( )] λ j,mj +1 Possible criterion for picking m j : (used in our Numerics) λ j,mj +1 < δ j H j H j... subdomain diameter, δ j... overlap V. Dolean Schwarz methods 102 / 107

103 Numerics Darcy I Domain & Partitions Coefficient Iterations (CG) vs. jumps Code: Matlab & FreeFem++ κ 2 ASM-1 ASM-2-low dim(v H ) GenEO dim(v H ) (8) 16 (8) (8) 17 (15) (8) 21 (15) (8) 18 (15) ASM-1: 1-level ASM ASM-2-low: m j = 1 NEW: λ j,mj +1 < δ j /H j V. Dolean Schwarz methods 103 / 107

104 Numerics Darcy II Iterations (CG) vs. number of subdomains regular partition subd. dofs ASM-1 ASM-2-low dim(v H ) GenEO dim(v H (4) 10 (6) (8) 11 (14) (16) 13 (30) > (32) 13 (62) METIS partition subd. dofs ASM-1 ASM-2-low dim(v H ) GenEO dim(v H (4) 15 (7) (8) 18 (15) (16) 22 (31) > (32) 34 (63) V. Dolean Schwarz methods 104 / 107

105 Numerics Darcy III Iterations (CG) vs. overlap (added) layers ASM-1 ASM-2-low (V H ) GenEO (V H ) (8) 11 (14) (8) 9 (14) (8) 9 (14) V. Dolean Schwarz methods 105 / 107

106 Numerics 2D Elasticity E IsoValue e e e e e e e e e e e e e e e e e e e e+11 E 1 = ν 1 = 0.3 E 2 = ν 2 = 0.45 METIS partitions with 2 layers added subd. dofs ASM-1 ASM-2-low (V H ) GenEO (V H ) (12) 42 (42) (48) 45 (159) (75) 47 (238) (192) 45 (519) V. Dolean Schwarz methods 106 / 107

107 Numerics 3D Elasticity Iterations (CG) vs. number of subdomains E 1 = ν 1 = 0.3 Relative error vs. iterations 16 regular subdomains E 2 = ν 2 = 0.45 subd. dofs ASM-1 ASM-2-low (V H ) GenEO (V H ) (24) 16 (46) (48) 16 (102) (96) 16 (214) V. Dolean Schwarz methods 107 / 107