AN INTRODUCTION TO DOMAIN DECOMPOSITION METHODS. Gérard MEURANT CEA

Transcription

1 Marrakech Jan 2003 AN INTRODUCTION TO DOMAIN DECOMPOSITION METHODS Gérard MEURANT CEA

2 Introduction Domain decomposition is a divide and conquer technique Natural framework to introduce parallelism in the solution of elliptic or parabolic PDE s General scheme: Decompose the problems into subproblems Solve the subproblems in parallel Glue the (sub)solutions together to get the global solution Generally the subproblems correspond to partitioning the domain The modern view on DD is to construct preconditioners for Krylov iterative methods for solving linear systems (CG, GMRES, BiCG,... depending on the properties of the matrix)

3 There are hundreds of variants of DD preconditioners Two main classes methods with subdomain overlapping (Schwarz like) methods without overlapping (interface problems) Methods may differ also on other issues: exact or inexact solvers for the subproblems solving a reduced system or the global system etc... Most DD methods for PDEs rely on mesh partitioning (in most cases we need a graph partitioner (METIS,...))

4 BooksonDD Proceedings of the DDM conferences (first in Paris 1987) SMITH, BJORSTAD and GROPP, Domain decomposition:parallel multilevel methods for elliptic partial differential equations, Cambridge University Press (1996) QUARTERONI and VALLI, Domain decomposition methods for partial differential equations, Clarendon Press (1999) MEURANT, Computer solution of large linear systems, North Holland (1999)

5 The classical Schwarz alternating method H.A. SCHWARZ, Uber einige abbildungsaufgaben, Ges. Math. Abh. vol 11, (1869) pp Solve a 2 nd order elliptic PDE with Dirichlet b.c. in a bounded 2D domain Ω The domain Ω is split into two (or more) overlapping subdomains Ω 1 and Ω 2 Γ i, i =1, 2, is the part of the boundary of Ω i enclosed in Ω Γ 1 Ω 2 Γ 2 Ω 1 Guess a value for the unknowns on the inner boundary Γ 1 Solve the problem exactly in Ω 1 Use the computed values on the inner boundary Γ 2 to solve exactly in Ω 2 Repeat the process until convergence

6 This very simple method almost always converges Can be analyzed at the continuous or discrete level We consider the Poisson model problem but the results are more generally true for self adjoint continuous bilinear forms Basic Schwarz = u = f in Ω, u Ω =0. u 2k = f in Ω 1, u 2k Γ1 = u 2k 1 Γ1, u 2k+1 = f in Ω 2, u 2k+1 Γ2 = u 2k Γ2, + given boundary conditions on the other parts of the boundary The bilinear form a of the problem is defined as a(u, v) = u vdx. The model problem can be written in variational form as Ω a(u, v) =(f,v), v H 1 0 (Ω).

7 Let V 1 = H 1 0 (Ω 1 ) and V 2 = H 1 0 (Ω 2 ) and the projectors P 1 and P 2 defined by a(p i v, w) =a(v, w), w V i,i=1, 2. Functions defined only on subdomains are extended by 0 to H 1 0 (Ω) a(u 2k u, v 1 )=0, v 1 V 1, u 2k u 2k 1 V 1, This gives a(u 2k+1 u, v 2 )=0, v 2 V 2, u 2k+1 u 2k V 2. u u 2k =(I P 1 )(u u 2k 1 ), Therefore, u u 2k+1 =(I P 2 )(u u 2k ). u u 2k+1 =(I P 2 )(I P 1 )(u u 2k 1 ).

8 This is known as the multiplicative Schwarz method We have to study the convergence of v 0 V, v 2k =(I P 1 )v 2k 1, v 2k+1 =(I P 2 )v 2k. Theorem (P.L. Lions, DDM 1987) If V = V 1 + V 2, where the overbar denotes the closure of the set, then v k 0. Moreover, if V = V 1 + V 2 then (I P 2 )(I P 1 ) c, c < 1.

9 The matrix form of the Schwarz alternating method Solve a 2 nd order elliptic equation in a rectangle using a 5 point FD scheme with the natural (rowwise) ordering. In block form A = D 1 B T 2 B 2 D 2 B T Suppose the mesh is partitioned as B m 1 D m 1 Bm T B m D m. Ω 2 Ω1

10 The matrix A (1) corresponding to Ω 1 is A (1) = D 1 B T 2 B 2 D 2 B T B p 2 D p 2 Bp 1 T B p 1 D p 1, The matrix A (2) corresponding to Ω 2 is A (2) = D l+1 B T l+2 B l+2 D l+2 B T l Let us denote the matrix A in block form as B m 1 D m 1 Bm T B m D m. A = ( ) A (1) A (1,2) X X and A = ( ) Y Y A (2,1) A (2), and let b 1 and b 2 be the restrictions of the right hand side b to Ω 1 and Ω 2 Note that A (1,2) has only one non zero block in the left lower corner and A (2,1) is zero except for the upper right block

11 We denote by x 1 and x 2 the unknowns in Ω 1 and Ω 2 We extend the vectors x 1 and x 2 to Ω by completing with the components of the previous iterate The Schwarz alternating method is A (1) x 2k 1 = b 1 + = B T p (x 2k 1 2 ) p, A(2) x 2k+1 2 = b 2 + B l+1 (x 2k 1 ) l x 2k 1 = x 2k 1 1 +(A (1) ) 1 (b 1 A (1) x 2k 1 1 A (1,2) x 2k 1 1,2 ), x 2k+1 2 = x 2k 2 +(A (2) ) 1 (b 2 A (2) x 2k 2 A (2,1) x 2k 2,1). ( ) x 2k = x 2k 1 (A + (1) ) 1 0 (b Ax 2k 1 ), 0 0 ( ) x 2k+1 = x 2k (A (2) ) 1 (b Ax 2k ).

12 By eliminating x 2k we obtain ( ) x 2k+1 = x 2k 1 (A +[ (1) ) ( ) (A (2) ) 1 A ( ) (A (2) ) 1 ( (A (1) ) ) ]r 2k 1, r 2k 1 = b Ax 2k 1. The Schwarz alternating method is nothing else than a preconditioned Richardson iteration This method can also be written with another notation We introduce restriction operators R 1 and R 2 x k 1 = R 1 x k, x k 2 = R 2 x k. In our example R 1 is simply ( I p 1 0) and R 2 =(0 I m l+1 ) A (1) = R 1 AR T 1, A (2) = R 2 AR T 2.

13 The first step of the iteration is: restriction by R 1 apply the inverse of R 1 AR1 T extension of the result by R1 T x 2k = x 2k 1 + R T 1 (R 1 AR T 1 ) 1 R 1 (b Ax 2k 1 ). The second step is x 2k+1 = x 2k + R2 T (R 2 AR2 T ) 1 R 2 (b Ax 2k ). Proposition The matrix P i = Ri T (R iari T ) 1 R i A, i =1, 2 is an orthogonal projection in the scalar product defined by A If ε k is the error, we have ε 2k =(I P 1 )ε 2k 1, ε 2k+1 =(I P 2 )ε 2k. P i is the discrete version of the projection operator we introduced earlier

14 The rate of convergence Let us consider a one dimensional Poisson model problem A (1) = of order p 1 and A (2) which is the same matrix but of order n l Proposition We have ε 2k i = i p ε2k p,i=1,...,p 1 ε 2k+1 i = n i +1 n l +1 ε2k+1 l,i= l +1,...,n.

15 This is because A (1) ε 2k 1.. ε 2k p 1 = ε 2k 1 p and ε 2k p = ε 2k 1 p, ε 2k+1 l = ε 2k l At the end of the first half step, the error is maximum for the node p and linear (being 0 at the ends of the interval) At the end of the second half step, the error is maximum for the node l and linear

16 Theorem At odd steps, the maximum of the (absolute value) of the error is obtained for node l and ε 2k+1 = l p n p +1 n l +1 ε2k 1. The larger the overlap (p l) the faster the convergence because n l +1=n p +1+(p l)

17 The same analysis can be done on this problem for a larger number of subdomains since the error is still linear on each subdomain The rate of convergence is slower when we have a large number of subdomains Number of iterations as a function of the overlap solid line: two subdomains, dashed: three subdomains

18 Other boundary conditions A way to reduce the overlap while maintaining a good convergence rate is to use other inner boundary conditions than Dirichlet for the subproblems (W.P. Tang, M. Gander and al.) WPT proposed using inner mixed boundary conditions like continuity of ωu +(1 ω) u n. Numerical results show that this can substantially improve the rate of convergence for small overlaps

19 Parallelizing Schwarz methods There is no parallelism in the Schwarz alternating method To get a parallel algorithm we may use a coloring of the subdomains such that a subdomain of one color is only connected to subdomains of other colors For strips a red black ordering is used, every other strip is black, and red strips alternate with black strips

20 The additive Schwarz method The alternating (multiplicative) Schwarz method can be considered as a kind of Gauss-Seidel algorithm A way to get a parallel algorithm is to use instead a block Jacobi like method This is known as the Additive Schwarz method, (Dryja and Widlund) x k+1 = x k + θ i R T i (R i AR T i ) 1 R i (b Ax k ) The preconditioner is M 1 = i R T i (R i AR T i ) 1 R i, Notice there is no parameter The iteration matrix is T AS = I θm 1 A where the summation is over the number of overlapping subdomains

21 More generally, one can replace the exact solves for each subdomain by approximations and define M 1 = i R T i M 1 i R i. Theory by Benzi, Frommer, Nabben and Szyld, Num. Math. v 89 (2001) pp see also Frommer and Szyld, Num. Math. v 83 (1999) pp Theorem (BFNS) A M matrix, θ<1/q, q max nb of ovl subdomains, then AS converges x 0, MS converges x 0 Theorem (Nabben) A M matrix, MS converges always faster than AS

22 Adding a coarse mesh correction The rate of convergence of the multiplicative or additive Schwarz methods depends on the number of subdomains To improve on this we add a coarse grid correction The coarse grid generally corresponds to the interfaces in the partitioning M 1 = i R T i (R i AR T i ) 1 R i + R T 0 A 1 0 R 0, The coarse grid operator may be chosen as a Galerkin approximation A 0 = R 0 AR0 T If the extent of overlap is kept proportional to the sizes of the subdomains the number of iterations is independent of n and of the number of subdomains

23 An additive Schwarz preconditioner for parabolic problems u t (a(x, y) u x x ) (b(x, y) u y y )=f in Ω R 2, u Ω =0, u(x, 0) = u 0 (x). The model problem that is considered is the heat equation : u t u = f, with Dirichlet boundary conditions, Ω being the unit square For stability and efficiency an implicit Crank Nicolson scheme We discretize the space variables with FD

24 Time is discretized with the usual C-N scheme :with t [0,T], k = t being the time step, p referring to the values of unknowns at time pk, we have u p+1 u p k + 1 2h 2 (Aup+1 + Au p )= 1 2 (f p+1 + f p ), At every time step this gives a linear system to solve: (2 h2 k I + A)up+1 =2 h2 k up Au p + h 2 (f p+1 + f p ). A t x = h 2 (f p+1 + f p ) 2Au p, where θ =2 h2 k, and A t = θi + A Then u p+1 = x + u p We remark that for our problem, A t is a symmetric strictly diagonally dominant M matrix

25 This method was inspired by an algorithm of Y. Kuznetsov It belongs to the class of Additive Schwarz methods Contrary to Kuznetsov, we use CG to solve the linear system at each time step and DD only to provide a preconditioner The domain Ω is divided into non overlapping subdomains Ω i, i =1,...,l Each Ω i is extended to a domain Ω i that overlaps the neighbors of Ω i (restricted of course to Ω)

26 To solve Mz = r, the following steps are performed : for each subdomain Ω i, i =1,...,l, let Âi be the n i n i matrix arising from the discretization of the problem on Ω i with homogeneous Dirichlet boundary conditions Let r i be the vector of length n i that will be the right hand side on Ω i, whose entries are equal to those of r for components corresponding to mesh points in Ω i and 0 elsewhere Then let ẑ i be defined by solving a problem on Ω i : Â i ẑ i = r i. We extend ẑ i to a vector z i on the whole Ω with 0 components corresponding to mesh points outside Ω i The solution z is simply defined as l z = ẑ i. i=1 The main problem is to know where to put the artificial boundaries

27 Numerical results Heat equation in the unit square, u = sin(πt)xy(1 x)(1 y)exp(xy), T =0.5, h =1/32, k = t =1/ Boxes: 2x nb of iterations overlapping 240 Boxes: 4x nb of iterations overlapping

28 350 Boxes: 8x nb of iterations overlapping

29 Algebraic domain decomposition methods without overlapping We consider a square domain Ω decomposed into two subdomains An elliptic second order PDE in a rectangle discretized by FD Let Ω 1 and Ω 2 be the two subdomains and Γ 1,2 the interface which is a mesh line Ω 2 Γ 1,2 Ω 1 We denote by m 1 (resp. m 2 ) the number of mesh lines in Ω 1 (resp. Ω 2 ), each mesh line having m mesh points (m = m 1 + m 2 +1)

30 We renumber the unknowns in Ω Let x 1 (resp. x 2 ) be the vector of unknowns in Ω 1 (resp. in Ω 2 )and x 1,2 be the vector of the unknowns on the interface A 1 0 E 1 0 A 2 E 2 E T 1 E T 2 A 12 x 1 x 2 x 1,2 = b 1 b 2. b 1,2 E 1 =( E m 1 1 ) T, E 2 =(E ) T, where E m 1 1 and E 1 2 are diagonal matrices Most algebraic DD methods are based on block Gaussian elimination (or approximate block Gaussian factorization) of the matrix Basically, we have two possibilities depending on the fact that we can or cannot (or do not want to) solve linear systems corresponding to subproblems like { A1 y 1 = c 1 A 2 y 2 = c 2 exactly with a direct method (or with a fast solver)

31 Exact solvers for the subdomains We eliminate the unknowns x 1 and x 2 in the subdomains This gives a reduced system for the interface unknowns Sx 1,2 = b 1,2, with b 1,2 = b 1,2 E T 1 A 1 1 b 1 E T 2 A 1 2 b 2 and S = A 12 E T 1 A 1 1 E 1 E T 2 A 1 2 E 2. The matrix S is the Schur complement of A 12 in A Constructing and factoring S is costly A more economical solution is to solve the reduced system with matrix S on the interface with an iterative method

32 Theorem For the Poisson model problem the condition number of the Schur complement is κ(s) =O( 1 h ) The product, Sp can be computed easily as Sp = A 1,2 p E T 1 A 1 1 E 1p E T 2 A 1 2 E 2p, p being a vector defined on the interface E 1 p =(0...0 E m 1 1 ) T p =(0...0 E m 1 1 p) T, E 2 p =(E ) T p =(E 1 2p 0...0) T. Then w 1 = A 1 1 E 1p is computed by solving A 1 w 1 = E 1 p, This is solving a linear system corresponding to a problem in Ω 1

33 Note that only the last block of the right hand side is different from 0 and because we only need E1 T w 1, the last block wm 1 1 solution w 1 is what we must compute of the Similarly, w 2 = A 1 2 E 2p is computed by solving A 2 w 2 = E 2 p, a problem in Ω 2 Finally, we have Sp = A 1,2 p wm 1 1 w1. 2 To improve the convergence rate of CG on the reduced system, a preconditioner M is needed The main problem is: Find an approximation of the Schur complement S

34 Schur complement, 2 subdomains

35 Approximate solvers for the subdomains Let us choose M in the form M = L M 1 1 M 1 2 M 1 1,2 L T, where M 1 (resp. M 2 ) is of the same order as A 1 (resp. A 2 )and M 1,2 is of the same order as A 1,2. L is block lower triangular L = M 1 0 M 2 E1 T E2 T M 1,2 At each PCG iteration, we must solve a linear system like Mz = M z 1 z 2 = r = r 1 r 2. z 1,2 r 1,2 This is done by first solving Ly = r, where the first parallel two steps are M 1 y 1 = r 1, M 2 y 2 = r 2.

36 Finally, we solve for the interface M 1,2 y 1,2 = r 1,2 E T 1 y 1 E T 2 y 2. To obtain the solution, we have a backward solve step as I 0 M 1 1 E 1 I M2 1 E z 1 2 z 2 = I z 1,2 This implies that z 1,2 = y 1,2 and y 1 y 2 y 1,2. M 1 w 1 = E 1 z 1,2, z 1 = y 1 w 1, M 2 w 2 = E 2 z 1,2, z 2 = y 2 w 2. How to choose the approximations M 1, M 2 and M 1,2?

37 where M = M 1 0 E 1 0 M 2 E 2 E T 1 E T 1 M 1,2, M 1,2 = M 1,2 + E T 1 M 1 1 E 1 + E T 2 M 1 2 E 2. We would like M to be an approximation of A, it makes sense to choose and M 1 A 1, M 2 A 2, M 1,2 A 1,2 = M 1,2 A 12 E T 1 M 1 1 E 1 E T 2 M 1 2 E 2. We are back to the same problem as before; that is to say, M 1,2 must be an approximation to the Schur complement S

38 Approximate Schur complements in the two subdomains case We suppose that A has a block tridiagonal structure and A = A 1 = A 2 = D 1 B T 2 B 2 D 2 B T B m 1 D m 1 Bm T B m D m D 1 B T 2 B 2 D 2 B T B m1 1 D m1 1 Bm T 1 B m1 D m1 D m1 +2 B T m 1 +3 B m1 +3 D m1 +3 B T m ,, B m 1 D m 1 Bm T B m D m, A 1,2 = D m1 +1, E m 1 1 = B T m 1 +1, E 1 2 = B m1 +2.

39 We consider a block twisted factorization of A, the block in the center being j = m A 1 =( +L 1 ) 1 ( + L T 1 ), where is a block diagonal matrix and L 1 is the block lower triangular part of A 1 which is of block order m 1 A 2 =(Σ+L T 2 )Σ 1 (Σ + L 2 ), where Σ is a block tridiagonal matrix and L 2 is the block lower triangular part of A 2 We denote the diagonal blocks of Σ by Σ m1 +2,...,Σ n { 1 = D 1, and i = D i B i 1 i 1 BT i, i =2,...,m 1 { Σm = D m, Σ i = D i B T i+1σ 1 i+1 B i+1, i = m 1,...,m 1 +2

40 Theorem S = D m1 +1 B m m 1 B T m 1 +1 B T m 1 +2Σ 1 m 1 +2 B m Now we consider the eigenvalues of the Schur complement for separable problems A = T I I T I I T I I T, T = QΛQ T, Q being such that QQ T = I and Λ being a diagonal matrix whose diagonal elements are the eigenvalues of T We denote these eigenvalues by λ l,l =1,...,m

41 Theorem The spectral decompositions of matrices i and Σ i are i = QΛ i Q T, Σ i = QΠ i Q T, i where Λ i and Π i are diagonal matrices whose diagonal elements are given for l =1,...,m by and (Λ 1 ) l,l =Λ l,l = λ l, (Λ i ) l,l = λ l 1 (Λ i 1 ) l,l, i =2,...,m 1 Proposition If λ l 2, then (Π m ) l,l = λ l, (Π i ) l,l = λ l 1 (Π i+1 ) l,l, i = m 1,...,m 1 +2 (Λ i ) l,l = (r l) i+1 + (r l ) i+1 (r l ) i + (r l ) i, i =1,...,m 1 (Π j ) l,l = (r l) m j+2 + (r l ) m j+2 (r l ) m j+1 + (r l ) m j+1, j = m,...,m 1 +2 where (r l ) ± = λ l± λ 2 l 4 2

42 Theorem The spectral decomposition of the Schur complement is S = QΘQ T, where Θ is a diagonal matrix whose diagonal elements θ l are given by θ l = λ l (r l) m 1 + (r l ) m 1 (r l ) m (r l ) m 1+1 where (r l ) ± = λ l± λ 2 l 4 2 (r l) m 2 + (r l ) m 2 (r l ) m (r l ) m 2+1, l =1,...,m We do not need to explicitly know the eigenvectors Q to compute the eigenvalues Proposition Let λ l =2+σ l and γ l = ( 1+ σ l 2 σ l + σ2 l 4 ) 2, then θ l = ( 1+γ m 1+1 l 1 γ m 1+1 l ) + 1+γm 2+1 l 1 γ m σ 2+1 l + σ2 l 4 l, l =1,...,m

43 We note that if we assume λ l > 2, l, as(r l ) ± > 0 and (r l ) + > (r l ), we have 0 <γ l < 1. Consider the case of the model problem λ l =2+σ l =4 2cos(lπh), l =1...,m where h = 1 m+1 σ min =2 2cos(πh) =2π 2 h 2 + O(h 4 ), σ max =2 2cos(π m m +1 )=4 O(h2 ). θ min = C 1 h + O(h 2 ), θ max = C 2 + O(h 2 ), where C 1 and C 2 are two constants independent of h and ( ) 1 κ(s) =O. h

44 Let us now look at the eigenvalues of S when, for a fixed h, the domains Ω 1 and Ω 2 extend to infinity Theorem If λ l > 2, θ l 2 σ l + σ2 l 4 when m i, i =1, 2. Let Σ 2 be the diagonal matrix of the eigenvalues σ i =2 2cos(iπh), i =1,...,m and Q such that q i,j = 2 m +1 sin(ijπh), i,j =1,...,m.

45 Dryja s preconditioner: Let T 2 be the matrix corresponding to finite difference discretization of the one dimensional Laplacian, T 2 = Q 2 Σ 2 Q T 2 M D = Q 2 Σ2 Q T 2. Golub and Mayers preconditioner: Theorem M GM = Q 2 Σ 2 + Σ2 2 4 QT 2. For the Poisson model problem κ(m 1 S)=O(1), for Dryja and Golub Mayers

46 The Neumann Dirichlet preconditioner This preconditioner was introduced by Bjørstad and Widlund A 1 0 E 1 0 A 2 E 2 E T 1 E T 2 A 1,2 x 1 x 2 x 1,2 = b 1 b 2, b 1,2 We can distinguish what in A 1,2 comes from subdomain Ω 1 and what comes from Ω 2 Since we know that A 1,2 = A (1) 1,2 + A(2) 1,2. we can define S = A 1,2 E T 1 A 1 1 E 1 E T 2 A 1 2 E 2, S (1) = A (1) 1,2 ET 1 A 1 1 E 1, S (2) = A (2) 1,2 ET 2 A 1 2 E 2, and S = S (1) + S (2)

47 The Neumann Dirichlet preconditioner is defined as M ND = S (1). Note, that we could also have chosen S (2) instead of S (1) Theorem For the Poisson model problem κ(m 1 ND S)=O(1). Note that if m 2 = m 1, the preconditioner is exact

48 Why is this preconditioner called Neumann Dirichlet? is equivalent to solving S (1) y 12 =(A (1) 1,2 ET 1 A 1 1 E 1)y 1,2 = c 1,2, ( )( ) ( ) A1 E 1 y1 0 E1 T A (1) =. 1,2 y 1,2 c 1,2 For second order elliptic PDEs, it is easy to see that this is simply solving a problem in Ω 1 with given Neumann boundary conditions on the interface When the solution is known on the interface, it is enough to solve a Dirichlet problem in Ω 2

49 The Neumann Neumann preconditioner This preconditioner was introduced by Le Tallec M 1 NN = 1 2 [(S (1) ) 1 +(S (2) ) 1] Note that we directly define the inverse of the preconditioner as an average of inverses of local (to each subdomain) inverses of Schur complements. Theorem For the Poisson model problem κ(m 1 NN S)=O(1).

50 Approximations of Schur complements with many subdomains We extend the results for two subdomains by considering now the domain Ω being divided into k strips We denote by x i the unknowns in subdomain Ω i and by x i,i+1 the unknowns on the interface between Ω i and Ω i+1

51 A = A 1 C 1 A 2 E 2 C 2 A 3 E C k 1 A k C T 1 E T 2 A 1,2 C T 2 E T 3 B 2, E k, C T k 1 E T k A k 1,k x = x 1 x 2 x 3.. x k x 1,2 x 2,3. x k 1,k, b = b 1 b 2 b 3.. b k b 1,2 b 2,3. b k 1,k.

52 A i = D 1 i (A 2 i )T A 2 i D 2 i (A 3 i )T A m i i D m i 1 i A m i i (A m i i D m i i ) T, i =1,...,k Proposition The Schur complement matrix S is block tridiagonal and we denote S = A 12 F T 2 F 2 A 23 F T F k 2 A k 2,k 1 F T k 1 F k 1 A k 1,k. A i,i+1 = A i,i+1 Ci T A 1 i C i Ei+1A T 1 i+1 E i+1, F i = C T i A 1 i E i.

53 Theorem The eigenvalues of S are ( 2 ω l,j = 1 γ m+1 σ l + σ2 l 1+γ m+1 l 2γ m+1 2 l 4 l l =1,...,m j =1,...k 1 cos ( )) jπ, k κ(s) =O ( ) k. h If k is fixed and h 0, κ increases as 1/h If h is given and we increase the number of subdomains, κ increases as k The problem is to define preconditioners for S

54 First idea:use block diagonal preconditioners, the diagonal blocks arising from the two subdomains case Ex:Dryja s; This preconditioner removes the h dependency but not that on k and, in fact, we have κ(m 1 D S)=O(k2 ). The same result is true for the Golub and Mayers preconditioner although the condition number is a little smaller It is more difficult to generalize the Neumann Dirichlet preconditioner to many subdomains If we do this in the obvious way, we have κ = O(k 2 ) The Neumann Neumann preconditioner can be easily extended to many subdomains, the inverses of partial Schur complements have to be weighted by the inverse of the number of subdomains which share a given node

55 Inexact subdomain solvers If we cannot solve exactly for the subproblems, we are not able to use an iterative method with S as we cannot compute the matrix vector product Sv We need a global parallel preconditioner M = L M 1 1 M M 1 k M 1 1,2... M 1 k 1,k L T

56 M 1 M L = M k C1 T E2 T M 1,2 C2 T E3 T H 2 M 2, C T k 1 E T k H k 1 M k 1,k The matrices M i can be chosen as for the two subdomains case For matrices M i,i+1 and H i, we have many possible choices

57 Domain decomposition with boxes A domain decomposition with strips can be done for more general domains by finding pseudo peripheral nodes and constructing the level structure corresponding to one of these nodes However, except for very large problems, when partitioning in this way, we cannot use many subdomains. A way to partition with many subdomains is to use so called boxes

58 Interfaces = two kinds of sets:edges and cross points We use an index E i for the edges, E = E i and V for the vertices Example:model problem with 121 nodes and 9 subdomains arranged as 3 3, each subdomain having 3 3 nodes

59 With this ordering the structure of S is given as nz = 540

60 S looks like

61 The Bramble, Pasciak and Schatz preconditioner a(u, v) =(f,v), v V where a is a coercive bilinear form arising from a second order elliptic PDE, V being a Hilbert space, say H0 1 (Ω) for homogeneous Dirichlet boundary conditions We want to construct another spectrally equivalent bilinear form b(u, v) such that λ 0 b(v, v) a(v, v) λ 1 b(v, v), v V and to use b as a preconditioner Ω is divided into non overlapping subdomains Ω k, the edges between two subdomains being denoted as Γ l,m Another intermediate form is introduced to eventually allow for some averaging of the coefficients, ã(u, v) = k a k i,j i,j Ω k u x i v x j dx = k ã k (u, v).

62 The method separates interior, edges and vertices unknowns in the following way: u = u P + u H, where u P is in V 0 (Ω k ) where functions in V 0 (Ω k ) have homogeneous Dirichlet boundary conditions and u P is defined by u P =0onΓ l,m. ã k (u P,φ)=ã k (u, φ), φ V 0 (Ω k ). This takes care of the right hand side and u H is defined by ã k (u H,φ)=0 φ V 0 (Ω k ).

63 The method goes one step further and decomposes u H interfaces as on the u H = u E + u V, where u E stands for edge unknowns, u V for vertices unknowns, u V (v j )=u(v j ) and u V Γij is linear, u E (v j )=0 BPS defined an operator l 0 on the edges: V 0 (Γ i,j ) V 0 (Γ i,j ) by c 1 l 0 (w)φ = cw φ, φ V 0 (Γ i,j ), Γ i,j Γ i,j where c is piecewise constant This defines something which behaves like the one dimensional Laplace operator b(w, φ) =ã(u P,φ P ) + α i,j c 1 l 1/2 0 (u E )φ E Γ Γ i,j i,j + Γ i,j(u V (v i ) u V (v j ))(φ V (v i ) φ V (v j )).

64 The basis functions that are used are the usual ones for the interior nodes, one dimensional hat functions for the edges (vanishing at the vertices) and functions which are linear on each edge, 1 at one vertex, 0 at the other ones for the vertices BPS requires us to perform the following steps: 1) solve Dirichlet problems on each subdomain in parallel for u P, 2) solve one dimensional edge equations in parallel for u E 3) solve a coarse mesh system on vertices for u V.Fromu E and u V we obtain the boundary values of u H 4) solve Dirichlet problems on each subdomain in parallel for u H. The solution is u P + u H Theorem Under suitable hypotheses, the condition number for the preconditioned system in two dimensions is κ C where H is the coarse mesh size. ( 1+log 2 ( H h )),

65 Variants of the BPS preconditioner can also be denoted as M 1 v = edges R T E i (α i M i ) 1 R Ei v + R T HA 1 H R Hv, where R Ei denotes the restriction to the edge E i and R H is a weighted restriction onto the coarse mesh, M i being one of the preconditioners for two subdomain case:either Dryja or Golub Mayers

66 Vertex space preconditioners A way to improve on BPS is to allow for some coupling between the vertices and the edge nodes Some points are considered around each vertex on each of the edges Let V k be this set of points. Then the preconditioner is defined as R T HA 1 H R Hv + edges M 1 v = R T E i (M Ei ) 1 R Ei v + vertices This includes some coupling between neighboring edges R T V k (M Vk ) 1 R Vk v The edge preconditioner can be chosen as a weighting of Dryja s or Golub Mayers preconditioners

67 The restriction to the edges is tridiagonal and an edge is only linked to the crosspoint and to the two nodes adjacent to the crosspoint on the neighboring edges If enough points are used around each vertex, then the condition number is independent of h and of the number of subdomains The vertex space algorithm was developed by B. Smith

68 In his Ph.D. thesis L. Carvalho considered some preconditioners whose spirit is quite close to the vertex space preconditioners Because they involve some kind of overlapping between the edge and vertex parts, they are denoted as algebraic additive Schwarz (AAS) He studied several local block preconditioners for the subdomains and several coarse space preconditioners For one of the local preconditioners, the main difference with the vertex space preconditioner is that the edge and the adjacent vertices are considered together * * * *

69 Another proposal was to consider the complete boundary of one subdomain, to be able to retrieve all the couplings between the edge nodes and the vertices when the interior nodes are eliminated It is necessary to add a coarse space component in the algorithm A restriction operator R 0 is defined (depending on the choice of the coarse part of the preconditioner) The coarse component of the preconditioner is defined as R T 0 A 1 0 R 0 where A 0 is the Galerkin coarse space operator A 0 = R 0 SR T 0

70 Several possibilities were considered: i) a subdomain based coarse space where all the boundary points of a subdomain are considered. The coarse space is spanned by vectors which have non zero components for the points around a subdomain, for all subdomains. ii) a vertex based coarse space where the vertices and some few adjacent edge points are considered. iii) an edge based coarse space where the points of an edge and the adjacent vertices are considered. When combining these coarse space preconditioners with the local parts, a preconditioner for which the condition number is insensitive to the mesh size or the number of subdomains is obtained except for very highly anisotropic problems

71 Numerical experiments mesh for each subdomain Pb 1:Poisson equation nb of subd M E M VE M S M C E M C VE M C S

72 Pb 2:Isotropic discontinuous pb on the Scottish flag, coefficients 1, 10 3, 10 3 nb of subd M C E M C VE M C S Pb 3:Anisotropic and discontinuous pb on the Scottish flag, coefficient 1 in x, same as before in y nb of subd M C E M C VE M C S