BALANCING DOMAIN DECOMPOSITION FOR PROBLEMS WITH LARGE JUMPS IN COEFFICIENTS

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1 MATHEMATICS OF COMPUTATION Volume 65, Number 216 October 1996, Pages BALANCING DOMAIN DECOMPOSITION FOR PROBLEMS WITH LARGE JUMPS IN COEFFICIENTS JAN MANDEL AND MARIAN BREZINA Abstract. Te Balancing Domain Decomposition algoritm uses in eac iteration solution of local problems on te subdomains coupled wit a coarse problem tat is used to propagate te error globally and to guarantee tat te possibly singular local problems are consistent. Te abstract teory introduced recently by te first-named autor is used to develop condition number bounds for conforming linear elements in two and tree dimensions. Te bounds are independent of arbitrary coefficient jumps between subdomains and of te number of subdomains, and grow only as te squared logaritm of te mes size. Computational experiments for two- and tree-dimensional problems confirm te teory. 1. Introduction Domain decomposition metods for solving elliptic boundary value problems ave received muc attention in te last few years. Te main reason for te popularity of tese metods is undoubtedly te need to take advantage of parallel computers, but many domain decomposition metods are efficient solvers in a classical uniprocessor environment as well. Tis paper is concerned wit a domain decomposition preconditioner for linear, conforming finite elements for te elliptic equation σ u) = fwit te coefficients σ canging between te subdomains by orders of magnitude. Te main component of te domain decomposition algoritms of te type studied ere is an approximate solver based on te solution of local independent subproblems on subdomains and a global coarse problem wit one or a few unknowns for eac subdomain to effect a global excange of information between te subdomain solution. Te composed approximate solver is ten used as a preconditioner in te conjugate gradients metod. It is well known tat te absence of a coarse problem results in deterioration of convergence of te iterations wit increasing number of subdomains [11, 14]. Te Balancing Domain Decomposition BDD) was introduced by Mandel [20] by adding a coarse problem to an earlier metod of De Roeck and Le Tallec [11], known as te Neumann-Neumann metod, based in turn on earlier work for te case of two subdomains [2] and on a closely related metod of Glowinski and Weeler for mixed Received by te editor Marc 18, 1993 and, in revised form, December 2, 1993 and September 21, Matematics Subject Classification. Primary 65N55, 65F10. Key words and prases. Domain decomposition, second-order elliptic boundary value problems, two-level iterative metods, discontinuous coefficients. Submitted Marc 1993; revised September c 1996 American Matematical Society

2 1388 JAN MANDEL AND MARIAN BREZINA problems [17]. Te development of BDD was motivated by very good performance of te Neumann-Neumann preconditioner for real-world problems wit strongly discontinuous coefficients for a small number of subdomains [11]. Algoritms similar to BDD but different in important aspects and convergence results also independent of coefficient jumps between subdomains were recently obtained by Dryja and Widlund [16], and by Sarkis [23]. For application of te BDD metod to mixed finite elements and computational results on a parallel computer, see Cowsar, Mandel, and Weeler [9]. For extensions of BDD to plate bending and performance results on a sared-memory parallel macine, see Le Tallec, Mandel, and Vidrascu [19]. In tis paper, we extend te abstract teory from [20] by an elementary argument sowing tat te convergence of te BDD metod is bounded independently on coefficient jumps of arbitrary size between subdomains. We obtain new bounds on convergence from te abstract teory by unifying te fundamental inequalities of te Domain Decomposition teory by Dryja and Widlund [12, 13, 26, 27] and Bramble, Pasciak, and Scatz [3, 5], and complementing tem wit some new results in te 2D case. In te 3D, we only need to use te inequalities from [5]. We sow tat te condition number after preconditioning is bounded by C1 + log 2 H/)), were H is te caracteristic subdomain size and is te caracteristic element size, bot in 2D and 3D. In te 3D case, suc a bound was already given in [20] based on a different estimate from [11]. Te teory is confirmed by computational experiments. Furter numerical results, available in te tecnical report [21], demonstrate tat te metod beaves very well even in te case of general discontinuities of te coefficients and irregular subdomain sapes in 2D and in many cases in 3D as well. Te paper is organized as follows: 2 introduces te BDD algoritm. Abstract bounds on te condition number are given in 3, relying only on algebraic arguments. Te assumptions of tese bounds are verified for finite element discretizations in 4. Finally, 5 contains numerical results. 2. Formulation of te problem and algoritms We will recall te notation and formulation of te algoritm, following [20]. Consider a system of linear algebraic equations 1) Ax = f, wit te m m, symmetric positive definite matrix A arising from a finite element discretization of a linear, elliptic, self-adjoint boundary value problem on a domain Ω. We assume te domain Ω to be split into nonoverlapping subdomains Ω 1,...,Ω k, eac of wic is te union of some of te elements. Let A i be te local stiffness matrix corresponding to subdomain Ω i, x i be te vector of degrees of freedom corresponding to all elements in Ω i,andletn i denote te matrix wit entries 0 or 1 mapping te degrees of freedom x i into global degrees of freedom, i.e., x i = Ni T x. Ten te stiffness matrix A is obtained by te standard subassembly process, A = N i A i Ni T. Eac x i is split into degrees of freedom x i tat correspond to Ω i, called interface degrees of freedom, and te remaining interior degrees of freedom ẋ i. Te degrees of freedom on Ω Ω i are assigned to te interiors. Te subdomain stiffness matrices

3 BALANCING DOMAIN DECOMPOSITION 1389 and te 0-1 matrices N i are ten split accordingly: ) ) xi Āi B i x i =, A ẋ i = i Bi T, N Ȧ i = N i,ṅ i ). i Assume te subdomain matrices A i to be symmetric and positive semidefinite and te submatrices Ȧi nonsingular. Witout loss of generality, let te interface degrees of freedom be numbered first and te interior degrees of freedom second in te global numbering. Let k Γ= Ω i, V i be te space of te interface degrees of freedom for te subdomain Ω i and V denote te space of all degrees of freedom on Γ, in a global numbering. After elimination of te interior degrees of freedom, te problem 1) reduces to a problem posedonteinterfacespacev, 2) Su = g, were S is te Scur complement 3) S = N i S i N T i, S i = Āi B i Ȧ 1 i B T i. Te local Scur complements S i are symmetric positive semidefinite and S is positive definite. Interpreting matrices as mappings, we ave 4) S : V V, S i : V i V i, Ni : V i V. Trougout tis paper, we denote u, v = u T v and, for symmetric positive semidefinite B, u, v B = Bu,v and u B = u, u B ) 1/2. Te notation u v means u, v =0. Muc of te benefit of domain decomposition is obtained already by solving te reduced problem 2) by conjugate gradients wit simple preconditioners suc as an approximation to te diagonal of S, cf. [6, 7, 18]. Evaluation of te action of S i can be implemented by solving a Diriclet problem on Ω i. Te BDD metod is based on te Neumann-Neumann preconditioner [11, 10], wic requires te solution of a Neumann problem on every subdomain Ω i named so in contrast to te Neumann-Diriclet preconditioner, wic requires solving Neumann problems on some subdomains and uses te original Diriclet problem on oters). An important design coice for te Neumann-Neumann preconditioner is te selection of weigt matrices D i tat form a decomposition of unity on te interface space V, T 5) N i D i N i = I. A straigtforward coice for D i is a diagonal matrix wit te diagonal elements being te reciprocal of te number of subdomains te degree of freedom is associated wit. A better coice, wic also guarantees a convergence bound independent of coefficient jumps between subdomains, is given in Teorem 3.3 below. For oter possibilities, see [11] and 5 below. Te following algoritm defines a linear operator M 1 N-N for use as a preconditioner.

4 1390 JAN MANDEL AND MARIAN BREZINA Algoritm 2.1 Neumann-Neumann preconditioner, [11]). Given r V, compute z = M 1 N-Nr as follows. Distribute r to subdomains, r i = Di T N i T r, solve te local problems 6) S i u i = r i on te subdomains, and average te results by z = N i D i u i. Since te matrices S i are typically singular, De Roeck and Le Tallec [11] used a pseudoinverse obtained by replacing zero pivots in te Gaussian decomposition by positive values. Te BDD metod adds a coarse problem as follows. Let n i =dimv i,0 m i n i,and Z i be n i m i matrices of full column rank suc tat 7) Ker S i Range Z i, i =1,...,k, and let W V be defined by W = {v V v = N i D i u i,u i Range Z i }. Te space W will play te role of a coarse space just as in variational multigrid metods [22]. We say tat s V is balanced if 8) Zi T DT i N i T s =0, i =1,...,k. Te process of replacing r by a balanced s = r Sw, w W, will be called balancing. We are now ready to define te action r z = M 1 u of te BDD preconditioner. Algoritm 2.2 BDD preconditioner, [20]). Given r V, compute M 1 r as follows. Balance te original residual by solving te auxiliary problem for unknown vectors λ i R mi, 9) and set 10) Z T i DT i s = r S N T i r S j=1 j=1 N j D j Z j λ j, N j D j Z j λ j ) =0, s i = D T i N T i s, i =1,...,k, i =1,...,k. Find any solution u i for eac of te local problems 11) S i u i = s i, i =1,...,k, balance te residual by solving te auxiliary problem for µ i R mi, 12) Z T i DT i N T i r S j=1 N j D j u j + Z j µ j ) ) =0, i =1,...,k,

5 BALANCING DOMAIN DECOMPOSITION 1391 and average te result on te interfaces according to 13) z = N i D i u i + Z i µ i ). If some m i =0,tenZ i as well as te block unknowns µ i and λ i are void and te it block equation is taken out of 9) and 12). Te presence of te coarse problem now guarantees tat te possibly singular local problems 11) are consistent, and te result of te algoritm does not depend on te coice of te solutions of 11), see [20]. In practice, te residual of te initial approximation sould be balanced first as in 12); ten te first balancing step 9) in every iteration can be omitted since te residual r received from te conjugate gradients algoritm is already balanced. 3. Algebraic teory In tis section, we give bounds on te condition number, relying on algebraic arguments only. Tese results apply to arbitrary linear systems of te form described in te preceding section. Teir assumptions will be verified in te following section for systems obtained from a particular variant of te Finite Element Metod. Te following teorem was proved in [20, Teorem 3.2] in te case wen Range Z i = Ker S i, but te same proof applies ere. Teorem 3.1. Algoritm 2.2 returns z = M 1 r,weremis symmetric positive definite and cond M,S) =λ max M 1 S)/λ min M 1 S) C, were { k j=1 C=sup N j T k N } i D i u i 2 S j k u u i V i, u i Ker S i, S i u i Range Z i. i 2 S i To motivate te bound given in Teorem 3.1, we need te concepts of glob and glob projection, defined as follows. Definition 3.2. Any vertex, edge, and, in te 3D case, face, of Γ will be called a glob. A glob is understood to be relatively open; for example, an edge does not contain its endpoints. We will also identify a glob wit te set of te degrees of freedom associated wit it. Te set of all globs will be denoted by G. For a glob G G, define te glob projection as follows: for a vector u V, E G u V is te vector tat as te same values as u for all degrees of freedom in G, andall oter degrees of freedom of E G u are zero. Te glob projection in terms of te local degrees of freedom is E ji G = N j TE G N i : V i V j. Note tat any two distinct globs from G are disjoint, and Γ = k Ω i = G G G. Te mappings E G, E ij G correspond to zero-one matrices and satisfy 14) E G = I, NT j Ni = E ji G, Eji G = Eji G Eii G, G G G G and 15) G Ω i Ω j E ji G 0, G Ω i EG ii 0. We are now ready for an abstract bound in te case wen te matrices S i are scaled by arbitrary positive numbers α i, wic corresponds to coefficient discontinuities of arbitrary size between te subdomains. Te teorem is formulated and proved in terms of properties of matrices only.

6 1392 JAN MANDEL AND MARIAN BREZINA Teorem 3.3. Let α i > 0, i =1,...,k, t 1/2,andE ji G, Ni, S i,andz i satisfy 3), 14), and7). Define D i as te diagonal matrices 16) D i = di, G)EG ii, di, G) = α t i G:E G 0 ii t j α j:e ji G 0 and assume tat tere exists a number R so tat for all i, j =1,...,k and all G, 1 17) E ji G α u i 2 S j 1 R u i 2 S j α i i for all u i suc tat u i Ker S i, S i u i Range Z i. Ten te weigt matrices D i form a decomposition of unity 5), and te preconditioner defined by Algoritm 2.2 satisfies 18) cond M,S) K 2 L 2 R, T were K =max i {j N j Ni 0}, andl=max i,j {G E ji G 0}. Proof. Te property 5) follows from te definition 16) and from 14), N i T D i N i = di, G)E G = E G = I. G:E G 0 ii G G Let j be fixed. Since tere are at most K nonzero terms in te sum k N j T N i D i u i, it follows by te triangle inequality and te Caucy inequality tat ) 2 N j T N i D i u i 2 S j N j T N i D i u i Sj K N j T N i D i u i 2 S j, and 19) j=1 N T j N i D i u i 2 S j K 2 max N j T j, N i D i u i 2 S j. If E ji G 0, te coefficient di, G) from 16) satisfies di, G) αt i /αt i + αt j ), and it follows from 14) and from 17) tat N T j N i D i u i Sj Now by 19), G:E ji G 0 α t i α t i + αt j E ji G u i Sj G:E ji G 0 LR 1/2 ρ 1/2 sup ρ>0 1+ρ t u i Si LR 1/2 u i Si. j=1 N T j N i D i u i 2 S j K 2 L 2 R u i 2 S i, α t 1/2 i α 1/2 j α t i + αt j R 1/2 u i Si wic concludes te proof, owing to Teorem 3.1. Note tat te constant K is te maximal number of adjacent subdomains Ω j to any subdomain Ω i plus one, and L is te maximal number of globs in any Ω i Ω j. If t>1/2, te estimate 18) can be sligtly improved; in particular, if t = 1, analogously to te metod of De Roeck and Le Tallec [11], one as cond M,S) K 2 L 2 R/2.

7 BALANCING DOMAIN DECOMPOSITION 1393 Te related metod of Dryja and Widlund [16] uses te coarse space W wit t =1/2 in 16), and te matrices S i in 11) replaced by S i + c i M i, M i positive definite, to avoid solving singular problems. Sarkis [23] obtained an estimate for a similar metod for nonconforming elements wit any t 1/2. 4. Teory for a finite element discretization Let Ω be a bounded domain in R d d =2ord= 3) wit a piecewise smoot boundary Ω, and Ω =Γ 1 Γ 2 wit Γ 1, Γ 2 disjoint, Γ 1 > 0. Consider te model problem 20) were 21) Lu = f in Ω, u = g on Γ 1, Lv = d r,s=1 x r u n =0onΓ 2, αx)β rs x) vx) ), x s wit te coefficient matrix β rs ) uniformly positive definite, bounded and piecewise smoot on Ω, and αx) a positive constant in eac subdomain Ω i, i.e., αx) =α i >0 for x Ω i. Let ˆΩ denote a reference domain of diameter O1) e.g., square or cube in 2D or 3D, respectively) and assume tat te subdomains Ω i are of diameter OH) and sape regular, i.e., 22) Ω i = F i ˆΩ), F i CH, Fi 1 CH 1, wit F i te Jacobian and te Euclidean R d matrix norm. Let V Ω) be a conforming linear finite element space on a triangulation of Ω suc tat eac subdomain Ω i is te union of some of te elements, and te usual sape regularity and inverse assumption old [8]. All functions v V Ω) satisfy omogeneous boundary condition u =0onΓ 1. Let V Ω i ) be te space of te restrictions of functions in V Ω) to Ω i. In all te estimates below, C and c denote generic positive constants independent of te sapeorsizeofωandω i. Note tat tese constants may depend on te constant in 22) or on te regularity of te triangulation, but tey are independent of and H. Following [4], [12] or [25], we define te scaled Sobolev norms u 2 1,Ω i = u 2 1,Ω i + 1 H 2 u 2 0,Ω i, u 2 1/2, Ω i = u 2 1/2, Ω i + 1 H u 2 0, Ω i, were u 2 1,Ω i = ux) 2 dx, u 2 ut) us) 1/2, Ω i = Ω i Ω i Ωi 2 t s d dtds. Te advantage of tis definition is tat it allows us to restrict all of our considerations to te reference domain ˆΩ and use te mappings F i to obtain te results for eac Ω i from te obvious norm equivalence c u 2 1,Ω i u F i 2 1,ˆΩ Hd 2 C u 2 1,Ω i, 23) c u 2 1/2, Ω i u F i 2 1/2, ˆΩ Hd 2 C u 2 1/2, Ω i.

8 1394 JAN MANDEL AND MARIAN BREZINA Assume tat for eac Ω i,γ 1 Ω i is eiter empty or a part of Ω i of size bounded below by a fixed proportion of te size of Ω i so tat te Poincaré inequality olds uniformly for all Ω i and wit te constant C independent of and H, 24) u 2 0,Ω i CH u 2 1,Ω i, u 2 0, Ω i CH 1/2 u 2 1/2, Ω i for all u V Ω i )ifγ 1 Ω i and for all u V Ω i ), Ω i uds = 0 if Γ 1 Ω i =. To apply Teorem 3.1, we first need to replace te S i norm by te scaled H 1/2 norm. Tis is a standard result [3, 13, 26], wic we state ere for reference in a form suitable for our purposes. Te scaling by α i is obvious. Lemma 4.1. Tere exist constants c>0,c independent of H or so tat c u 2 1/2, Ω i 1 α i u 2 S i C u 2 1/2, Ω i, u V Ω i ). To derive te fundamental inequality 17) assumed in Teorem 3.3, we identify by abuse of notation) V wit V Γ) and V i wit V Ω i ). Ten te glob projections are E G : V Γ) V Γ), and 17) becomes a bound on te increase of te H 1/2 norm wen a function in V Ω i ) is canged by setting its values to zero on all nodes of Ω i \ G. We first consider te two-dimensional case, Ω R 2.Since Ω i is one-dimensional, we may use te properties of te space V 0,H) of piecewise linear functions on a uniform mes wit step on te interval [0,H]. Te following form of Discrete Sobolev Inequality was proved by Dryja [12]. Lemma 4.2. Tere exists a constant C suc tat u 2 L 0,H) C 1+log H ) u 2 H 1/2 0,H), u V 0,H). We will also need te following bound for te H 1/2 norm of te extension by zero from an interval to te wole R, proved by Bramble, Pasciak, and Scatz [3, Lemma 3.5]. Lemma 4.3. Tere exists a constant C suc tat for all u V 0,H) satisfying u0) = uh) =0,u=0outside 0,H), u 2 1/2,R C 1+log H ) u 2 L 0,H) + u 2 1/2,0,H). An estimate of te H 1/2 norm of a spike function, obtained by sampling te value of a given function at one point, follows easily. Lemma 4.4. Tere exists a constant C suc tat for all u V 0,H),0 1, and v 0 V R) defined by v 0 0) = u0), v 0 x)=0for x, v 0 2 1/2,R C 1+log H ) u 2 1/2,0,H). Proof. Let L = u L 0,H). It follows from Lemma 4.3 tat v 0 2 1/2,R C 1+log 2 ) 25) v 0 2 L,) + v 0 2 1/2,,).

9 BALANCING DOMAIN DECOMPOSITION 1395 Using linearity of v 0,weobtain v 0 2 1/2,,) = v 0 s) v 0 t) 2 26) s t 2 dsdt 4 L 2, because v 0 2 L,) = v 0 0) 2 L 2. Tus, v 0 2 1/2,,) CL2. But L 2 C1 + log H ) u 2 1/2,0,H) by Lemma 4.2, wic concludes te proof. By subtracting suc spikes at te endpoints, we can extend Lemma 4.3 to te case wen te values of u at te endpoints are nonzero. Lemma 4.5. Tere exists a constant C so tat for u V 0,H) and w V R) suc tat w = u on [, H ], and wx) =0for x 0, x H, w 2 1/2,R C 1+log H ) 2 u 2 1/2,0,H). Proof. Define ux) to be zero for x, ) H +, ), and linear in [, 0] and [H, H + ]. Furter, define v 0 and v H by v 0 x) = { u0), x =0, 0, x, v 0 linear in [, 0] and in [0,], { uh), x = H, v H x) = 0, x H, v H linear in [H, H] andin[h, H + ]. Writing w as w = u v 0 v H,and applying Lemma 4.3 and Lemma 4.4, we obtain w 2 1/2,R C 1+log H ) w 2 L 0,H) + w 2 1/2,0,H) = C 1+log H ) u 2 L 0,H) + w 2 1/2,0,H) C 1+log H ) u 2 L 0,H) +3 u 2 1/2,0,H) + v 0 2 1/2,R + v H 2 ) 1/2,R C 1+log H ) ) u 2 L 0,H) + u 2 1/2,0,H) +1+logH ) u 2 1/2,0,H). Application of Lemma 4.2 to u L,0,H) concludes te proof. We are now ready for te estimate of te H 1/2 norm of te glob projections E G, wic sows tat an arbitrary function in V Ω i ) can be decomposed into its glob parts wit only a small increase in te H 1/2 norm. Teorem 4.6. Let Ω R d, d =2or d =3. Ten tere exists a constant C not dependent of or H, so tat for any glob G G and for all u V Ω i ), E G u 2 1/2, Ω i C 1+log H ) 2 u 2 1/2,G. Proof. In te 2D case, te proposition follows by using a mapping of Ω i onto an interval so tat G maps to 0,H), from Lemma 4.5 for G being an edge, and from Lemma 4.4 for G being a vertex.

10 1396 JAN MANDEL AND MARIAN BREZINA In te 3D case, te proposition was proved for te case of G beingafaceof Ω i as Lemma 4.3 in [5]. In te case of G being an edge or a vertex of Ω i, te proof follows from Lemma 4.2. and te proof of Lemma 4.1 of [5]. Te bound on te condition number of te BDD algoritm follows. Teorem 4.7. Let Ω R d, d =2or d =3, and te weigt matrices D i be diagonal wit te entries given by 16). Ten tere exists a constant C independent of H, and α i, so tat te condition number of te BDD metod satisfies cond M,S) C 1+log 2 H Proof. We only need to verify te assumption 17) of Teorem 3.3. Lemma 4.1 allows to replace te S i norms by te H 1/2 Ω i ) seminorms, wic may in turn be replaced by te H 1/2 Ω i ) norms, owing to te Poincaré inequality 24). It remains to use Teorem 4.6. ). 5. Computational results Te purpose of our computational tests was to demonstrate te fast convergence of te BDD metod on complicated problems wit varying coefficients. In all of te following examples, te space V of te piecewise linear functions defined on a uniform rectangular mes of stepsize in 2D or 3D was used for te solution of te elliptic problem of te form 20), u divσ u) =1inΩ, u =1onΓ 1, n =0onΓ 2, wit Ω =Γ 1 Γ 2,Γ 1 Γ 2 =. Te coefficient σ is an elementwise constant function, and k is te number of subdomains. We ave compared tree algoritms: conjugate gradients applied to te reduced system 2) witout preconditioning denoted as CG in te tables), conjugate gradients wit Neumann-Neumann preconditioner and te local singular problems 6) solved using te Moore-Penrose pseudoinverse Algoritm 2.1, denoted as N-N), and conjugate gradients wit te BDD preconditioner using Range Z i = Ker S i Algoritm 2.2, denoted as BDD). Te stopping criterion for te iterations was based on 27) λ max M 1 S) λ min M 1 S) M 1 r, r M 1 b, b ɛ2, wit r te current residual and b te rigt-and side, wic guarantees te relative precision of ɛ in te energy norm, cf. Asby, Manteuffel, and Saylor [1]. Te condition number λ max M 1 S)/λ min M 1 S) reported in te tables and also used in 27) was estimated as te ratio of te extreme Ritz values for te Krylov space, computed from te eigenvalues of a tridiagonal matrix constructed from te Lanczos recursion in conjugate gradients. Number of iterations wit means tat te criterion 27) was not satisfied wen te maximum number of iterations was reaced. Te 2D examples were computed by a prototype implementation of te BDD metod programmed using te CLAM package [24]. In te two-dimensional test BDD implementation, te weigts D i were based on te diagonal entries of te Scur complements, as suggested in [11], because we ad te diagonal entries of te

11 BALANCING DOMAIN DECOMPOSITION 1397 Table 1. 2D results for Poisson equation on unit square Fig. 1, σ 1 = σ 2 =1) CG N-N BDD k iterations cond. iterations cond. iterations cond. 1/ / , / , Table 2. 2D ceckerboard pattern Fig. 1, σ 1 =10 3,σ 2 =10 3 ) CG N-N BDD k σ 1 σ 2 iter cond. iter cond. iter cond. 1/ / * / , Table 3. 2 x 2 ceckerboard pattern for various σ 1,σ 2 CG N-N BDD k σ 1 σ 2 iter cond. iter cond. iter cond. 1/ * / * , / * , Scur complement available: Denote s i ll te diagonal entry of Scur complement S i corresponding to global degree of freedom l. For subdomain Ω i, te weigt matrix D i was constructed as diagonal wit diagonal elements d i ψ il),wereψ il)iste local number in Ω i associated wit te global degree of freedom l, d i ψ il) = s i ll j:l Ω j s j ll wic is essentially 16) computed node by node wit te diagonal entries of S i used instead of te scalars α i. Tis coice of te weigts was found to give good results [11]. Te domain Ω was cosen to be te unit square and Γ 1 was te left-and side of Ω. Te tests sow tat unlike for te CG and N-N metod, te condition number and te number of iterations of te BDD metod does not deteriorate for increasing number of subdomains Table 1, Fig. 1), te coefficient σ varying by orders of magnitude between te subdomains Table 2, Fig. 1), and increasing jumps in te coefficients Table 3).,

12 1398 JAN MANDEL AND MARIAN BREZINA Γ 2 σ 1 σ 2 σ 1 σ 2 σ 1 σ 2 σ 1 σ 2 σ 1 σ 2 Γ σ 1 σ 2 σ 1 σ 2 σ 1 1 Γ 2 σ 2 σ 1 σ 2 σ 1 σ 2 σ 1 σ 2 σ 1 σ 2 σ 1 Γ 2 Figure 1. 2D ceckerboard pattern Table 4. 3D Poisson s equation, various and number of subdomains x y z k dof iter cond. 1/15 1/15 1/ /20 1/25 1/ /30 1/30 1/ A FORTRAN implementation was used for te 3D experiments, wit te action of S i implemented in a straigtforward way following te definition of te Scur complement 3). Te implementation of te action of te inverse, tat is, te solution of S i y = x, relies on te obvious fact tat y may equivalently be computed, using notation of 2, as solution of te sparse system ) ) ) Āi B i y x =, z 0 B T i Ȧ i

13 BALANCING DOMAIN DECOMPOSITION 1399 Table 5. 3D ceckerboard pattern wit alternating σ k σ 1 σ 2 dof iter cond. 1/ / / / / / / / discarding z afterwards. Since te diagonal entries of S i are not available, te weigts were defined from α i = σ i by 16) wit t = 1. Te problem was set on unit cube Ω, wit zero Diriclet boundary condition on te wole Ω, and ɛ =10 18 was used for te stopping criterion 27). Again, te results confirm te teory. Finally, one sould note tat te l 2 norm of residual of te global solution was never larger tan 20 times te l 2 residual of te reduced solution. For furter numerical results, see [21]. FORTRAN 77 code tat implements te metod is available from MGNET by anonymous ftp to casper.cs.yale.edu in te directory /mgnet/jmandel. Te code invokes user-supplied subroutines tat implement te matrix-vector multiplications S i x i and solution of te possibly singular systems S i z i = r i. Acknowledgements Te autors are grateful to Olof Widlund for inspiring discussions and for providing in is papers te available teoretical tools in a systematic form tat made te simple proofs presented ere possible, and in particular for making available to us early drafts of [15] as well as oter publications. We also tank Lawrence Cowsar for useful discussions and pointing out several improvements in an earlier version of te paper. Tis researc was supported by NSF grants DMS , ASC , and ASC References 1. S. Asby, T. A. Manteuffel, and P. E. Saylor, A taxonomy for conjugate gradient metods, SIAM J. Numer. Anal., ), pp MR 91i: J.-F. Bourgat, R. Glowinski, P. Le Tallec, and M. Vidrascu, Variational formulation and algoritm for trace operator in domain decomposition calculations, in Domain Decomposition Metods, T. Can, R. Glowinski, J. Périaux, and O. Widlund, eds., SIAM, Piladelpia, PA, MR 90b: J. H. Bramble, J. E. Pasciak, and A. H. Scatz, Te construction of preconditioners for elliptic problems by substructuring, I, Mat. Comp., ), pp MR 87m: , An iterative metod for elliptic problems on regions partitioned into substructures, Mat. Comp., ), pp MR 88a: , Te construction of preconditioners for elliptic problems by substructuring, IV, Mat. Comp., ), pp MR 89m: T. F. Can, Analysis of preconditioners for domain decomposition, SIAM J. Numer. Anal., ), pp MR 88e:65033

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15 BALANCING DOMAIN DECOMPOSITION O. B. Widlund, An extension teorem for finite element spaces wit tree applications, in Numerical Tecniques in Continuum Mecanics, W. Hackbusc and K. Witsc, eds., Braunscweig/Wiesbaden, 1987, Notes on Numerical Fluid Mecanics, v. 16, Friedr. Vieweg und Son, pp Proceedings of te Second GAMM-Seminar, Kiel, January, , Iterative substructuring metods: Algoritms and teory for elliptic problems in te plane, in First International Symposium on Domain Decomposition Metods for Partial Differential Equations, R. Glowinski, G. H. Golub, G. A. Meurant, and J. Périaux, eds., SIAM, Piladelpia, PA, MR 90c:65138 Center for Computational Matematics, University of Colorado at Denver, Denver, Colorado address: address: