partial dierential equations by iteratively solving subproblems dened on and localized treatment of complex and irregular geometries, singularities

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1 Acta Numerca (1994), pp. 61{143 Doman decomposton algorthms Tony F. Chan Department of Mathematcs, Unversty of Calforna at Los Angeles, Los Angeles, CA 90024, USA Emal: Tarek P. Mathew Department of Mathematcs, Unversty of Wyomng, Larame, WY , USA Emal: Doman decomposton refers to dvde and conquer technques for solvng partal derental equatons by teratvely solvng subproblems dened on smaller subdomans. The prncpal advantages nclude enhancement of parallelsm and localzed treatment of complex and rregular geometres, sngulartes and anomalous regons. Addtonally, doman decomposton can sometmes reduce the computatonal complexty of the underlyng soluton method. In ths artcle, we survey teratve doman decomposton technques that have been developed n recent years for solvng several knds of partal dfferental equatons, ncludng ellptc, parabolc, and derental systems such as the Stokes problem and mxed formulatons of ellptc problems. We focus on descrbng the salent features of the algorthms and descrbe them usng easy to understand matrx notaton. In the case of ellptc problems, we also provde an ntroducton to the convergence theory, whch requres some knowledge of nte element spaces and elementary functonal analyss. The authors were supported n part by the Natonal Scence Foundaton under grant ASC , by the Army Research Oce under contract DAAL03-91-G-0150 and subcontract under DAAL03-91-C-0047, and by the Oce for Naval Research under contract ONR N J-1890.

2 62 T.F. Chan and T.P. Mathew CONTENTS 1 Introducton 62 2 Overlappng subdoman algorthms 70 3 Nonoverlappng subdoman algorthms 74 4 Introducton to the convergence theory 91 5 Some practcal mplementaton ssues Multlevel algorthms Algorthms for locally rened grds Doman mbeddng or cttous doman methods Convecton{duson problems Parabolc problems Mxed nte elements and the Stokes problem Other topcs 128 References Introducton Doman decomposton (DD) methods are technques for solvng partal dfferental equatons based on a decomposton of the spatal doman of the problem nto several subdomans. Such reformulatons are usually motvated by the need to create solvers whch are easly parallelzed on coarse gran parallel computers, though sometmes they can also reduce the complexty of solvers on sequental computers. These technques can often be appled drectly to the partal derental equatons, but they are of most nterest when appled to dscretzatons of the derental equatons (ether by - nte derence, nte element, spectral or spectral element methods). The prmary technque conssts of solvng subproblems on varous subdomans, whle enforcng sutable contnuty requrements between adjacent subproblems, tll the local solutons converge (wthn a speced accuracy) to the true soluton. In ths artcle, we focus on descrbng teratve doman decomposton algorthms, partcularly on the formulaton of precondtoners for soluton by conjugate gradent type methods. Though many fast drect doman decomposton solvers have been developed n the engneerng lterature, see Kron (1953) and Przemeneck (1963) (these are often called substructurng or tearng methods), the more recent developments have been based on the teratve approach, whch s potentally more ecent n both tme and storage. The earlest known teratve doman decomposton technque was proposed n the poneerng work of H. A. Schwarz n 1870 to prove the exstence of harmonc functons on rregular regons whch are the unon of overlappng subregons. Varants of Schwarz's method were later studed by Sobolev (1936), Morgenstern (1956) and Babuska (1957). See also Courant

3 Doman decomposton survey 63 and Hlbert (1962). The recent nterest n doman decomposton was ntated n studes by Dnh, Glownsk and Peraux (1984), Dryja (1984), Golub and Mayers (1984), Bramble, Pascak and Schatz (1986b), Bjrstad and Wdlund (1986), Lons (1988), Agoshkov and Lebedev (1985) and Marchuk, Kuznetsov and Matsokn (1986), where the prmary motvaton was the nherent parallelsm of these methods. There are not many general references that provde an overvew of the eld, but here are a few: dscussons n Keyes and Gropp (1987), Canuto, Hussan, Quarteron and Zang (1988), Xu (1992a), Dryja and Wdlund (1990), Hackbusch (1993), Le Tallec (1994) and the books of Lebedev (1986), Kang (1987) and Lu, Shh and Lem (1992) and the forthcomng book by Smth, Bjrstad and Gropp (1994). The best source of references remans the collecton of conference proceedngs: Glownsk, Golub, Meurant and Peraux (1988), Chan, Glownsk, Peraux and Wdlund (1989, 1990), Glownsk, Kuznetsov, Meurant, Peraux and Wdlund (1991), Chan, Keyes, Meurant, Scroggs and Vogt (1992a), Quarteron (1993). Ths artcle s conceptually organzed n three parts. The rst part (Sectons 1 through 5) deals wth second-order self-adjont ellptc problems. The algorthms and theory are most mature for ths class of problem and the topcs here are treated n more depth than n the rest of the artcle. Most doman decomposton methods can be classed as ether an overlappng or a nonoverlappng subdoman approach, whch we shall dscuss n Sectons 2 and 3 respectvely. A basc theoretcal framework for studyng the convergence rates wll be summarzed n Secton 4. Some practcal mplementaton ssues wll be dscussed n Secton 5. The second part (Sectons 6{8) consders algorthms that are not, strctly speakng, doman decomposton methods, but that can be studed by the general framework set up n the rst part. The key dea here s to extend the concept of the subdomans to that of subspaces. The topcs nclude multlevel precondtoners (Secton 6), locally rened grds (Secton 7) and cttous doman methods (Secton 8). In the last part (Sectons 9{12), we consder doman decomposton methods for more general problems, ncludng convecton{duson problems (Secton 9), parabolc problems (Secton 10), mxed nte element methods and the Stokes problems (Secton 11). In Secton 12, we provde references to algorthms for the bharmonc problem, spectral element methods, ndente problems and nonconformng nte element methods. Due to space lmtaton, and the fact that both the theory and algorthms are generally less well developed for these problems, we do not treat Parts II and III n as much depth as n Part I. Our am s nstead to hghlght some of the key deas, usng the framework and termnology developed n Part I, and to provde a gude to the vast developng lterature. We present the methods n algorthmc form, expressed n matrx notaton, n the hope of makng the artcle accessble to a broad spectrum of readers.

4 64 T.F. Chan and T.P. Mathew Gven the space lmtaton, most of the theorems (especally those n Parts II and III) are stated wthout proofs, wth ponters to the lterature gven nstead. We also do not cover nonlnear problems or specc applcatons (e.g. CFD) of doman decomposton algorthms. In the rest of ths secton, we ntroduce the man features of doman decomposton procedures by descrbng several algorthms based on the smpler case of two subdoman decomposton for solvng the followng general second-order self-adjont, coercve ellptc problem: Lu r (a(x; y)ru) = f(x; y); n ; u = 0 (1.1) We are partcularly nterested n the soluton of ts dscretzaton (by ether nte elements or nte derences) whch yelds a large sparse symmetrc postve dente lnear system: 1.1. Overlappng subdoman approach Au = f: (1:2) Overlappng doman decomposton algorthms are based on a decomposton of the doman nto a number of overlappng subregons. Here, we consder the case of two overlappng subregons f^ 1 ; ^ 2 g whch form a coverng of ; see Fgure 1. We shall let ; = 1; 2 denote the part of the boundary of whch s n the nteror of. The basc Schwarz alternatng algorthm to solve (1.1) starts wth any sutable ntal guess u 0 and constructs a sequence of mproved approxmatons u 1 ; u 2 ; : : :: Startng wth the kth terate u k, we solve the followng two subproblems on ^ 1 and ^ 2 successvely wth the most current values as boundary condton on the artcal nteror boundares: and 8 > < >: 8 >< >: Lu k+1 u k+1 u k+1 Lu k+1 The terate u k+1 s then dened by 1 = f; on ^ 1 ; 1 = u k j 1 on 1 ; 1 = 0; ^ 1 n 1 ; 2 = f; on ^ 2 ; u k+1 2 = u k+1 1 j 2 on 2 ; u k+1 2 = 0; ^ 2 n 2 : u k+1 (x; y) = ( u k+1 2 (x; y) f (x; y) 2 ^ 2 u k+1 1 (x; y) f (x; y) 2 n^ 2 : It can be shown that n the norm nduced by the operator L, the terates fu k g converge geometrcally to the true soluton u on,.e. ku u k k k ku u 0 k;

5 Doman decomposton survey 65 Nonoverlappng subdomans ^ ^2 1 = 1 [ 2 = ^ 1 [ ^ 2, ^ 1 \ ^ 2 6= ; Fg. 1. Two subdoman decompostons. where < 1 depends on the choce of ^ 1 and ^ 2. The above Schwarz procedure extends almost verbatm to dscretzatons of (1.1). We shall descrbe the dscrete algorthm n matrx notaton. Correspondng to the subregons f^ 1 ; ^ 2 g, let f^i 1 ; ^I 2 g denote the ndces of the nodes n the nteror of doman ^ 1 and nteror of ^ 2 respectvely. Thus ^I1 and ^I2 form an overlappng set of ndces for the unknown vector u. Let ^n 1 be the number of ndces n ^I1, and let ^n 2 be the number of ndces n ^I2. Due to overlap, ^n 1 + ^n 2 > n, where n s the number of unknowns n. Correspondng to each regon ^, we dene a rectangular n ^n extenson matrx R T whose acton extends by zero a vector of nodal values n ^. Thus, gven a subvector x of length ^n wth nodal values at the nteror nodes on ^ we dene: (R T x ) k = ( (x ) k for k 2 ^I 0 for k 2 I ^I ; where I = ^I1 [ ^I2 : The entres of the matrx R T are ones or zeros. The transpose R of ths extenson map R T s a restrcton matrx whose acton restrcts a full vector x of length n to a vector of sze ^n by choosng the entres wth ndces ^I correspondng to the nteror nodes n ^. Thus, R x s the subvector

6 66 T.F. Chan and T.P. Mathew of nodal values of x n the nteror of ^. The local subdoman matrces (correspondng to the dscretzaton on ^ ) are, therefore, A 1 = R 1 AR T 1 ; A 2 = R 2 AR T 2 ; and these are prncpal submatrces of A. The dscrete verson of the Schwarz alternatng method, descrbed earler, to solve Au = f, starts wth any sutable ntal guess u 0 and generates a sequence of terates u 0 ; u 1 ; : : : as follows u k+1=2 = u k + R T 1 A 1 1 R 1(f Au k ); (1.3) u k+1 = u k+1=2 + R T 2 A 1 2 R 2(f Au k+1=2 ): (1.4) Note that ths corresponds to a generalzaton of the block Gauss{Sedel teraton (wth overlappng blocks) for solvng (1.1). At each teraton, two subdoman solvers are requred (A 1 1 and A 1 2 ). Denng P R T A 1 R A; = 1; 2; the convergence s governed by the teraton matrx (I P 2 )(I P 1 ), hence ths s often called a multplcatve Schwarz teraton. Wth sucent overlap, t can be proved that the above algorthm converges wth a rate ndependent of the mesh sze h (unlke the classcal block Gauss{Sedel teraton). We note that P 1 and P 2 are symmetrc wth respect to the A nner product (see Secton 4), but not so for the teraton matrx (I P 2 )(I P 1 ). A symmetrzed verson can be constructed by teratng one more half-step wth A 1 1 after equaton (1.4). The resultng teraton matrx becomes (I P 1 )(I P 2 )(I P 1 ) whch s symmetrc wth respect to the A nner product and therefore conjugate gradent acceleraton can be appled. An analogous block Jacob verson can also be dened: u k+1=2 = u k + R T 1 A 1 1 R 1(f Au k ); (1.5) u k+1 = u k+1=2 + R T 2 A 1 2 R 2(f Au k ): (1.6) Ths verson s more parallelzable because the two subdoman solves can be carred out concurrently. Note that by elmnatng u k+1=2, we obtan u k+1 = u k + (R T 1 A 1 1 R 1 + R T 2 A 1 2 R 2)(f Au k ): Ths s smply a Rchardson teraton on Au = f wth the followng addtve Schwarz precondtoner for A: M 1 as = R T 1 A 1 1 R 1 + R T 2 A 1 2 R 2: The precondtoned system can be wrtten as M 1 as A = P 1 + P 2 ; whch s symmetrc wth respect to the A nner product and can also be used

7 Doman decomposton survey 67 wth conjugate gradent acceleraton. Agan, for sutably chosen overlap (see Secton 1), the condton number of the precondtoned system s bounded ndependently of h (unlke classcal block Jacob) Nonoverlappng subdoman approach Nonoverlappng doman decomposton algorthms are based on a partton of the doman nto varous nonoverlappng subregons. Here, we consder a model partton of nto two nonoverlappng subregons 1 and 2, see Fgure 1, wth nterface B 1 \@ (separatng the two regons). Let u = (u 1 ; u 2 ; u B ) denote the soluton u restrcted to 1, 2 and B respectvely. Then, u 1, u 2 satsfy the followng local problems: 8 < : Lu 1 = f n 1 u 1 = 0 1 nb u 1 = u B on B and 8 < : Lu 2 = f n 2 u 2 = 0 2 nb u 2 = u B on B (1:7) as well as the followng transmsson boundary condton on the contnuty of the ux across B: n 1 (aru 1 ) = n 2 (aru 2 ) on B; where each n s the outward pontng normal vector to B from. (We omt dervaton of the above, but note that t can be obtaned by applyng ntegraton by parts to the weak form of the problem.) Thus, f the value u B of the soluton u on B s known, the local solutons u 1 and u 2 can be obtaned at the cost of solvng two subproblems on 1 and 2 n parallel. The man task n nonoverlappng doman decomposton s to determne the nterface data u B. To ths end, an equaton satsed by u B can be obtaned by usng the transmsson boundary condtons. Let g denote arbtrary Drchlet boundary data on B. Dene E 1 g and E 2 g as solutons of the followng local problems, on 1 and 2 respectvely: 8 < : L(E 1 g) = f n 1 E 1 g = 0 1 nb E 1 g = g on B and 8 < : L(E 2 g) = f n 2 E 2 g = 0 2 nb E 2 g = g on B: (1:8) Then, by constructon the boundary values of E 1 g and E 2 g match on B (and equal g). However, n general the ux of the two local solutons wll not match on B,.e. n 1 (are 1 g) 6= n 2 (are 2 g) on B; unless g = u B. Dene the followng ane lnear mappng T whch maps the boundary data g on B to the jump n the ux across B: T : g! n 1 (are 1 g) + n 2 (are 2 g) :

8 68 T.F. Chan and T.P. Mathew Thus, the boundary value u B of the true soluton u, satses the equaton T u B = 0: (1:9) The map T s referred to as a Steklov{Poncare operator, and s a pseudoderental operator (Agoshkov, 1988; Quarteron and Vall, 1990). A property of the map T (or a lnear map derved from T snce t s ane lnear) s that t s symmetrc, and postve dente wth respect to the L 2 nner product on B. The dscrete versons of system (1.9) can therefore be solved by precondtoned conjugate gradent methods. We now consder the correspondng algorthm for solvng the lnear system Au = f. Based on the partton = 1 [ 2 [ B, let I = I 1 [ I 2 [ I 3 denote a partton of the ndces n the lnear system, where I 1 and I 2 conssts of the ndces of nodes n the nteror of 1 and 2, respectvely, whle I 3 conssts of the nodes on the nterface B. Correspondngly, the unknowns u can be parttoned as u = [u 1 ; u 2 ; u 3 ] T and f = [f 1 ; f 2 ; f 3 ] T, and the lnear system (1.2) takes the followng block form: 2 4 A 11 0 A 13 0 A 22 A 23 A T 13 A T 23 A u 1 u 2 u = 4 f 1 f 2 f : (1:10) Here, the blocks A 12 and A 21 are zero only under the assumpton that the nodes n 1 are not drectly coupled to the nodes n 2 (except through nodes on B), and ths assumpton holds true for nte element and loworder nte derence dscretzatons. As n the contnuous case, the problem Au = f can be reduced to an equvalent system for the unknowns u 3 on the nterface B. If u 3 s known, then u 1 and u 2 can be determned by usng the rst two block rows of (1.10): u 1 = A 1 11 (f 1 A 13 u 3 ) and u 2 = A 1 22 (f 2 A 23 u 3 ) : Substtutng for u 1 and u 2 n the thrd block row of (1.10), we obtan a reduced problem for the unknowns u 3 : Su 3 = f ~ 3 ; (1:11) where S A 33 A T 13 A 1 11 A 13 A T 23 A 1 22 A 23 and f ~ 3 f 3 A T 13 A 1 11 f 1 A T 23 A 1 22 f 2. The matrx S s referred to as the Schur complement of A 33 n A, and the equaton Su 3 f ~ 3 = 0 s a dscrete approxmaton of the Steklov{Poncare equaton T u B = 0, enforcng the transmsson boundary condton. The Schur complement S also plays a key role n the followng block LU factorzaton of (1.10) 2 4 I I 0 A T 13 A 1 11 A T 23 A 1 22 I A 11 0 A 13 0 A 22 A S u 1 u 2 u = 4 f 1 f 2 f ; (1:12)

9 Doman decomposton survey 69 from whch (1.11) can also be derved. Solvng (1.11) by drect methods can be expensve snce the Schur complement S s dense and, moreover, computng t requres as many solves of each A system as there are nodes on B. Therefore, t s common practce to solve the Schur complement system teratvely va precondtoned conjugate gradent methods. Each matrx{ vector multplcaton wth S nvolves two subdoman solvers (A 1 12 and A 1 22 ) whch can be performed n parallel. It can be shown that the condton number of S s O(h 1 ) (whch s better than that of A but can stll be large) and therefore a good precondtoner s needed. Note that an advantage of the nonoverlappng approach over the overlappng approach s that the terates are shorter vectors Man features of doman decomposton algorthms The two precedng algorthms extend naturally to the case of many subdomans. However, a straghtforward extenson wll not be scalable,.e. the convergence rate wll deterorate as the number of subdomans ncrease. Ths s necessarly so because n the above algorthms, the only mechansm for sharng nformaton s local,.e. ether through the nterface or the overlappng regons. However, for ellptc problems the doman of dependence s global (.e. the Green functon s nonzero throughout the doman) and some way of transmttng global nformaton s needed to make the algorthms scalable. One of the most commonly used mechansms s to use coarse spaces, e.g. solvng an approprate problem on a coarser grd. Ths wll be descrbed n detal later. In ths sense, many of the doman decomposton algorthms can be vewed as a two-scale procedure,.e. there s a ne grd wth sze h on whch the soluton s sought and on whch the subdoman problems are solved, as well as a coarse grd wth mesh sze H whch provdes the global couplng between dstant subdomans. The goal s to desgn the approprate nteracton of these two mechansms so that the resultng algorthm has a convergence rate that s as nsenstve to h and H as possble. In fact, n the lterature on doman decomposton, a method s called optmal f ts convergence rate s ndependent of h and H. In practce, however, an optmal precondtoner does not necessarly provde the least executon tme or mnmal computatonal complexty. To acheve a computatonally ecent algorthm requres payng attenton to other factors, n addton to h and H. Frst of all, even though the number of teratons requred by an optmal method can be bounded ndependent of h and H, one stll has to ensure that t s not large. Second, each teraton step must not cost too much to mplement. In addton, t would be desrable for the convergence rate to be nsenstve to the varatons n the coecents

10 70 T.F. Chan and T.P. Mathew of the ellptc problem, as well as the aspect ratos of the subdomans. We shall touch on some of these ssues later. We summarze here the key features of doman decomposton algorthms that we have ntroduced n ths secton, and whch we shall study n some detal n the rest of ths artcle: 1 doman decomposton as precondtoners wth conjugate gradent acceleraton; 2 overlappng versus nonoverlappng subdoman algorthms; 3 nonoverlappng algorthms nvolve solvng a Schur complement system, usng nterface precondtoners; 4 addtve versus multplcatve algorthms; 5 optmal precondtoners requre solvng a coarse problem; 6 the goal of achevng a convergence rate and ecency ndependent of h, H, coecents and geometry. Notaton We use the notaton cond (M 1 A) to denote the condton number of the precondtoned system M 1=2 AM 1=2, where M s symmetrc and postve dente. We call a precondtoner M spectrally equvalent to A f cond (M 1 A) s bounded ndependently of the mesh szes h and H, whchever s approprate. 2. Overlappng subdoman algorthms We now descrbe Schwarz algorthms based on many overlappng subregons to solve (1.1). We rst dscuss a commonly used technque for constructng an overlappng decomposton of nto p subregons ^ 1 ; : : :; ^ p. To ths end, let 1 ; : : :; p denote a nonoverlappng partton of. For nstance, each subregon may be chosen as elements from a coarse nte element trangulaton H of of mesh sze H. Next, we extend each nonoverlappng regon to ^, consstng of all ponts n wthn a dstance of H from where ranges from 0 to 0(1). See Fgure 2 for an llustraton of a two-dmensonal rectangular regon parttoned nto sxteen overlappng subregons. Once the extended subdomans ^ are dened, we dene restrcton maps R, extenson maps R T, and local matrces A correspondng to each subregon ^ as follows. Let A be nn and let ^n be the number of nteror nodes n ^. For each = 1; : : :; p, let ^I denote the ndces of the nodes lyng n the nteror of ^. Thus f^i1 ; : : :; ^Ip g form an overlappng collecton of ndex sets. For each regon ^ let R denote the n ^n restrcton matrx (whose entres consst of 1s and 0s) that restrcts a vector x of length n to R x of length ^n, by choosng the subvector havng ndces n ^I (correspondng to the nteror nodes n ^ ). The transpose R T of R s referred to as an extenson or nterpolaton matrx, and t extends subvectors of length ^n on

11 Doman decomposton survey ^ 1 ^ 3 ^ 2 ^ 4 ^ 9 ^ 11 ^ 10 ^ 12 Colour 1 Colour 2 ^ 5 ^ 7 ^ 6 ^ 8 ^ 13 ^ 15 ^ 14 ^ 16 Colour 3 Colour 4 Fg. 2. Nonoverlappng subdomans, overlappng subdomans ^, 4 colours.

12 72 T.F. Chan and T.P. Mathew ^ to vectors of length n usng extenson by zero to the rest of. Fnally, we let A = R AR T, whch s the local stness matrx correspondng to the subdoman ^. Snce R and R T have entres of 1's and 0's, each A s a prncpal submatrx of A Addtve Schwarz algorthms The most straghtforward generalzaton of the two subdoman addtve Schwarz precondtoners descrbed n Secton 1 to the many subdoman case s the followng: M 1 as;1 = px =1 R T A 1 R : Snce the acton of each term R T A 1 R z can be computed on separate processors, ths mmedately leads to coarse gran parallelsm. The actons of R T and R are scatter{gather operatons, respectvely, and t s not necessary to store the extenson and restrcton matrces. The precondtoner M as;1 s a straghtforward generalzaton of the standard block Jacob precondtoner to nclude overlappng blocks. However, the algorthm s not scalable because the convergence rate of ths precondtoned teraton deterorates as the number of subdomans p ncreases (.e. as H decreases). Theorem 1 There exsts a postve constant C ndependent of H and h (but possbly dependent on the coecents a) such that: Proof. cond (M 1 as;1 A) CH : See Dryja and Wdlund (1992a; 1989b). Ths deteroraton n the convergence rate can be removed at a small cost by ntroducng a mechansm for global communcaton of nformaton. There are several possble technques for ths, and here we wll descrbe the most commonly used mechansm whch s sutable only when the ne grd h s a renement of the coarse mesh H. Accordngly, let R T H denote the standard nterpolaton map of coarse grd functons to ne grd functons (as n two-level multgrd methods). In the nte element context, R T H smply nterpolates the nodal values from the coarse grd vertces to all the vertces on the ne grd, say by pecewse lnear nterpolaton. Its transpose R H s thus a weghted restrcton map. If there are n c coarse grd nteror vertces, then R T H wll be an n n c matrx. Indeed, f 1 ; : : :; n c are n c column vectors representng the coarse grd nodal bass functons on the ne grd, then R T H = 1; : : :; n c :

13 Doman decomposton survey 73 Correspondng to the coarse grd trangulaton H, let A H denote the coarse grd dscretzaton of the ellptc problem,.e. A H = R H AR T H. Then, the mproved addtve Schwarz precondtoner M as;2 s dened by M 1 as;2 = RT H A 1 H R H + px =1 R T A 1 R = px =0 R T A 1 R ; (2:1) where we have let R 0 = R H and A 0 = A H. The convergence rate usng ths precondtoner s ndependent of H (for sucent overlap). Theorem 2 There exsts a postve constant C ndependent of H, h (but possbly dependent on the varaton n the coecents a) such that cond (M 1 as;2 A) C : Proof. See Dryja and Wdlund (1992a; 1989b), Dryja, Smth and Wdlund (1993) and Theorems 14 and 16 n Secton Multplcatve Schwarz algorthms The multplcatve Schwarz algorthm for many overlappng subregons can be analogously dened. Startng wth an terate u k, we compute u k+1 as follows u k+(+1)=(p+1) = u k+=(p+1) + R T A 1 R (f Au k+=(p+1) ); = 0; 1; : : :; p: Theorem 3 The error ku u k k n the kth terate of the above multplcatve Schwarz algorthm satses ku u k k k ku u 0 k; where < 1 s ndependent of h and H, and depends only on and the coecents a, and k k s the A-norm. Proof. 16. See Bramble, Pascak, Wang and Xu (1991) and Theorems 15 and As for the addtve Schwarz algorthm, f the coarse grd correcton s dropped, then the convergence rate of the multplcatve algorthm wll deterorate as O(H 2 ) when H! 0. The multplcatve algorthm as stated above has less parallelsm than the addtve verson. However, ths can be mproved through the technque of multcolourng, as follows. Each subdoman s dented wth a colour such that subdomans of the same colour are dsjont. The multplcatve Schwarz algorthm then terates sequentally through the derent colours, but now all the subdoman systems of the same colour can be solved n parallel. Typcally, only a small number of colours s needed, see Fgure 2 for an example. We cauton that the convergence rate of the multcoloured

14 74 T.F. Chan and T.P. Mathew algorthm can depend on the orderng of the subdomans n the teraton and the ncreased parallelsm may result n slower convergence (well known for the classcal pontwse Gauss{Sedel method). However, ths eect s less notceable when a coarse grd solve s used. The convergence bounds we have stated for both the addtve and multplcatve Schwarz algorthms are vald n both two and three dmensons, but wth possble dependence on the varaton n the coecents a. For large jumps n the coecents, the convergence rate can deterorate, but wth maxmum possble deteroraton stated below. Theorem 4 Assume that the coecents a are constant (or mldly varyng) wthn each coarse grd element. Then, for the addtve Schwarz algorthm n two dmensons, and n three dmensons, cond (Mas;2 1 A) C (1 + log(h=h)) ; cond (Mas;2 1 A) C (H=h) ; where C s ndependent of the jumps n the coecents and the mesh parameters H and h, but dependent on the overlap parameter. Proof. See Dryja and Wdlund (1987) and Dryja et al. (1993). Correspondng results exst for the multplcatve Schwarz algorthms and the deteroraton n the convergence rate can be mproved by the use of alternatve coarse spaces, see precedng reference. For a numercal study of Schwarz methods, see Gropp and Smth (1992). 3. Nonoverlappng subdoman algorthms As we saw n Secton 2, there are two knds of couplng mechansms present n an optmal Schwarz type algorthm based on many overlappng subregons: local couplng between adjacent subdomans provded by the overlapped regons, and global couplng between dstant subdomans provded by the coarse grd problem. In the case of nonoverlappng approach, the Schur complement system represents the couplng between the nodes on the nterface B and n order to obtan optmal convergence rates, a coarse grd solve s stll needed. However, snce there s no overlap between neghbourng subdomans, the local couplng must be provded by some other mechansm. The most often used method s to use nterface precondtoners,.e. an effectve approxmaton to the part of the Schur complement matrx S that corresponds to the unknowns on the nterface separatng two neghbourng subdomans. (In two dmensons, the nterface s an edge and n three dmensons t s a face.) We shall rst descrbe such nterface precondtoners n Secton 3.1 n the context of two subdoman decomposton (where t s

15 Doman decomposton survey 75 the only precondtoner needed). The case of many subregons s dscussed n Secton Two nonoverlappng subdomans: nterface precondtoners Consder the same settng as n Secton 1, wth parttoned nto two subdomans 1 and 2 separated by an nterface B. We need a precondtoner M for the Schur complement S A 33 A T 13 A 1 11 A 13 A T 23 A 1 22 A 23: (1) Exact egen-decomposton of S: In some specal cases, an exact egen-decomposton of S can be derved from whch the acton of S 1 can be computed ecently. For example, consder the ve-pont dscretzaton of on a unform grd of sze h on the rectangular doman = [0; 1] [0; l 1 + l 2 ], whch s parttoned nto two subdomans 1 = [0; 1] [0; l 1 ] and 2 = [0; 1] [l 1 ; l 1 + l 2 ] wth nterface B = f(x; y) : y = l 1 ; 0 < x < 1g: We assume that the grd s n (m m 2 ) wth l = (m + 1)h, for = 1; 2 and h = 1=(n + 1). It was shown by Bjrstad and Wdlund (1986) and Chan (1987) that S = F F; where F s the orthogonal sne transform matrx: (F ) j = s 2 n + 1 sn j n + 1 s a dagonal matrx wth elements gven by () = 1 + m m 1+1 where = 4 sn 2 2(n + 1) and m m 2+1 ; = (1 + =2! q + 2 =4; q + 2 =4)2 : If m 1 ; m 2 are large enough, then two good approxmatons to S are: M GM = F ( + 2 =4) 1=2 F; and M D = F 1=2 F; where = dag( ). M D was rst used by Dryja (1982) n a more general settng. The mproved precondtoner M GM was later proposed by Golub and Mayers (1984). Note that all the above precondtoners can be solved n O (n log(n)) operatons usng the Fast Sne Transform and t s easy to show that they are spectrally equvalent to S. In theory, ths s true for any second-order ellptc operator. However, these precondtoners can be senstve to the aspect ratos l 1 and l 2 and the coecents (n the case of varable coecents) on the subdomans. To apply ths class of precondtoners to domans more general

16 76 T.F. Chan and T.P. Mathew than a rectangle, and to provde some adaptvty to aspect ratos, Chan and Resasco (1985; 1987) suggested usng the exact egen-decomposton of a rectangle whch approxmates the gven doman and shares the same nterface. Exact egen-decompostons have also been derved by Resasco (1990) for three-dmensonal problems and unequal mesh szes n each subdoman, and by Chan and Hou (1991) for ve pont stencls approxmatng general second-order constant coecent ellptc problems (whch provdes some adaptvty to the coecents). (2) The Neumann{Drchlet precondtoner (See Bjrstad and Wdlund (1984), Bjrstad and Wdlund (1986), Bramble et al. (1986b), Marn and Quarteron (1989).) To descrbe ths method, t s convenent to rst wrte S n a form whch reects the contrbutons from 1 and 2 more explctly. In ether nte derence or nte element methods, the term A 33 can be wrtten as A 33 = A (1) 33 + A(2) 33 ; where A () 33 corresponds to the contrbuton to A 33 from subdoman (assumng the coecents are zero on the adjacent subdoman). For nstance, n the case of nte elements, A () 33 s obtaned by ntegratng the weak form on. We can now wrte where S = S (1) + S (2) ; S () = A () 33 AT 3 A 1 A 3; = 1; 2: Due to symmetry, S (1) = S (2) = 1 2S f the two subdoman problems are symmetrc about the nterface. Ths motvates the use of ether S (1) or S (2) as a precondtoner for S even f the two subdomans are not equal. For example, a rght-precondtoned system usng M ND = S (1) has the form (S (1) + S (2) )S (1) 1 = I + S (2) S (1) 1 : It can be shown that the acton of S (1) 1 on a vector v can be obtaned by solvng a problem on 1 wth v as Neumann boundary condton on the nterface and extractng the soluton values (Drchlet values) on the nterface: S (1) 1 v = 0 I " A 11 A 13 A T 13 A (1) 33 # 1 0 v It s proved n Bjrstad and Wdlund (1986) that ths precondtoner s spectrally equvalent to S. (3) The Neumann{Neumann precondtoner One may notce a lack of symmetry n the Neumann{Drchlet precondtoner n the choce of whch subdoman to solve the Neumann problem on. The Neumann{Neumann :

17 Doman decomposton survey 77 precondtoner, rst proposed by Bourgat, Glownsk, Le Tallec and Vdrascu (1989), s completely symmetrc wth respect to the two subdomans. Here the nverse of the precondtoner s gven by M 1 NN = 1 4 S(1) S(2) 1 : Obvously, the acton of M 1 NN requres solvng a Neumann problem on each of the two subdomans. In addton to the added symmetry, ths precondtoner s also more drectly generalzable to the case of many subdomans and to three dmensons (see Sectons 3.6 and 3.9). (4) Probng precondtoner Ths purely algebrac technque, rst proposed by Chan and Resasco (1985) and later rened n Keyes and Gropp (1987) and Chan and Mathew (1992), s motvated by the observaton that the entres of the rows (and columns) of the matrx S often decay rapdly away from the man dagonal. Ths decay s faster than the decay of the Green functon of the orgnal ellptc operator. The dea n the probng precondtoner s to ecently compute a banded approxmaton to S. Note that ths would be easy f S was known explctly because we could then smply take the central dagonals of S. However, recall that we want to avod computng S explctly. The technque used n probng s to nd such an approxmaton by probng the acton of S on a few carefully selected vectors. For example, f S were trdagonal, then t can be exactly recovered by ts acton on the three vectors: v 1 = (1; 0; 0; 1; 0; 0; : : :) T ; v 2 = (0; 1; 0; 0; 1; 0; : : :) T ; v 3 = (0; 0; 1; 0; 0; 1; : : :) T through a smple recurson. Snce S s not exactly trdagonal, the trdagonal matrx M P obtaned by probng wll not be equal to S, but t s often a very good precondtoner. Keyes and Gropp (1987) showed that f S were symmetrc, then two probng vectors suce to compute a symmetrc trdagonal approxmaton. For more detals, see Chan and Mathew (1992), where t s proved that the condtoner number of M 1 P S can be bounded by O(h 1=2 ) (hence M P s not spectrally equvalent to S) but t adapts very well to the aspect ratos and the coecent varatons of the subdomans. It would seem deal to combne the advantages of the probng technque wth a spectrally equvalent technque but ths has proved to be elusve. (5) Multlevel precondtoners These technques make use of the multlevel ellptc precondtoners to be dscussed n Secton 6 and adapt them to obtan precondtoners for the Schur complement nterface system. We wll not descrbe these methods n detal, but the man dea s smple to understand. If a change of bass from the standard nodal bass to a herarchcal nodal bass s used (assumng that the grd has a herarchcal structure),

18 78 T.F. Chan and T.P. Mathew then a dagonal scalng often provdes an eectve precondtoner n the new bass. It can be shown rather easly that the Schur complement of the matrx A n the herarchcal bass s the same as that obtaned by representng S wth respect to the herarchcal bass on the nterface B (.e. by a multlevel change of bass restrcted to the nterface). Thus a good multlevel precondtoner for A automatcally leads to a good multlevel precondtoner for S. The reader s referred to Smth and Wdlund (1990) for usng the herarchcal bass method of Yserentant (1986) and Tong, Chan and Kuo (1991) (see also Xu (1989)) for the multlevel nodal bass method of Bramble, Pascak and Xu (1990). The resultng methods have optmal or almost optmal convergence rates Many nonoverlappng subdomans Many of the precondtoners descrbed n Secton 3.1 for two nonoverlappng subdomans can be extended to the case of many nonoverlappng subregons. However, n the case of many subregons, these precondtoners need to be moded to take account of the more complex geometry of the nterface, and to provde global couplng amongst the many subregons. Let be parttoned nto p nonoverlappng regons of sze O(H) wth nterface B separatng them, see Fgure 3: = 1 [ [ p [ B; where \ j = ; for 6= j; the nterface B s gven by: B = f[ p g \ : For = 1; : : :; p, let I denote the ndces correspondng to the nodes n the nteror of subdoman, and let I = [ p =1 I denote the ndces all nodes lyng n the nteror of subdomans. To mnmze notaton, we wll use B to denote not only the nterface, but also the ndces of the nodes lyng on B. Then, correspondng to the permuted ndces fi; Bg, the vector u can be parttoned as u = [u I ; u B ] T, and f = [f I ; f B ] T, and equaton (1.2) can be wrtten n block form as follows AII A IB ui fi = : (3.1) u B f B A T IB A BB For ve-pont stencls n two dmensons and seven-pont stencls n three dmensons, A II wll be block dagonal, snce the nteror nodes n each subdoman wll be decoupled from the nteror nodes n other subdomans: A II = blockdag (A ) = A A pp : (3.2) As n Secton 1, the unknowns u I can be elmnated resultng n a reduced system for u B (the unknowns on B). We use the followng block LU

19 Doman decomposton survey vertex (x H k ; yh k ) V m a vertex subregon j an edge E j factorzaton of A: A AII A T IB Fg. 3. A partton of nto 12 subdomans. A IB A BB = " I 0 I A T IB A 1 II # AII 0 0 S where the Schur complement matrx S s dened by S = A BB A T IBA 1 II A IB: " I A 1 II A IB 0 I # ; (3:3) Consequently, solvng Au = f based on the LU factorzaton above requres computng the acton of A 1 II twce, and S 1 once. By elmnatng u I, we obtan Su B = ~ f S ; (3:4) where f ~ B f B A IB A 1 II f I. The Schur complement S n the case of many subdomans has smlar propertes to the two subdoman case. Here we only note that the condton number of S s approxmately O(H 1 h 1 ) n the case of many subdomans, an mprovement over the O(h 2 ) growth for A. The rest of ths secton wll be devoted to the descrpton of varous precondtoners M for S n two and three dmensons Two-dmensonal case: block Jacob precondtoner M 1 For S Here, we descrbe a block dagonal precondtoner M 1 whch reduces the condton number of S from O(H 1 h 1 ) to O H 2 log 2 (H=h) (wthout nvolvng global communcaton of nformaton). A varant of ths precondtoner was proposed by Bramble, Pascak and Schatz (1986a), see also Wdlund (1988), Dryja et al. (1993).

20 80 T.F. Chan and T.P. Mathew The precondtoner M 1 wll correspond to an addtve Schwarz precondtoner for S correspondng to a partton of the nterface B nto subregons. The nterface B s parttoned as a unon of edges E for = 1; : : :; m, and vertces V of the subdomans, see Fgure 3: B = fe 1 [ [ E m g [ V; where the edges E j l form the common boundary of two subdomans (excludng the endponts). Wth duplcty of notaton, we also denote by E the ndces of the nodes lyng on edge E, and use V to denote the ndces of the vertces V. Correspondng to ths orderng of ndces, we partton u B = [u E1 ; : : :; u Em ; u V ], and obtan a block partton of S: S = S E1 E 1 S E1 E 2 S E1 E m S E1 V SE T 1 E 2 S E2 E 2 S E2 E m S E2 V SE T 1 E m SE T 2 E m S EmEm S EmV S T E 1 V S T E 2 V S T E mv S V V Note that S E E j = 0 f E and E j are not part of the same subdoman. A block dagonal (Jacob) precondtoner for S s: M 1 = S E1 E S E2 E SEmEm S V V The precondtoner M 1 can also be descrbed n terms of restrcton and extenson maps. For each edge E, let R E denote the pontwse restrcton map from B onto the nodes on E, and let R T E denote the correspondng extenson map. Smlarly, let R V denote the pontwse restrcton map onto the vertces V, and let R T V denote extenson by zero of nodal values on V to B. Then the block Jacob precondtoner s dened by M 1 1 mx =1 R T E S 1 E E R E + R T V S 1 V V R V : Snce ths precondtoner does not nvolve global couplng between subdomans, ts convergence rate deterorates as H! 0. Theorem 5 There exsts a constant C ndependent of H and h (but may depend on the coecent a), such that cond (M 1 1 S) CH log 2 (H=h) : : :

21 Doman decomposton survey 81 Proof. See Bramble et al. (1986a), Wdlund (1988), Dryja et al. (1993). Snce the S E E s are not explctly constructed, computng the acton of S 1 E E poses a problem (smlarly for S V V ). Fortunately, each S E E and S V V can be replaced by ecent approxmatons. For example, the block entres S E E can be replaced by any sutable two subdoman nterface precondtoner M E E dscussed n Secton 3.1, for nstance: M E E E F 1=2 F; where E represents the average of the coecent a n the two subdomans adjacent to E. Alternatvely, the acton of SE 1 E can be computed exactly, usng S 1 E E z E = 0 0 I A 1 j [ k [E 0 0 ze T ; (3:5) where E j \@ k, and A j [ k [E s the 33 block parttoned stness matrx correspondng to the regon j [ k [ E. Note that ths nvolves solvng a problem on j [ k [ E. The matrx S V V may be approxmated by the dagonal matrx A V V (the prncpal submatrx of A correspondng to nodes on V ) Two-dmensonal case: the Bramble{Pascak{Schatz (BPS) precondtoner M 2 for S The H 2 factor n the condton number of the block Jacob precondtoner M 1 can be removed by ncorporatng some mechansm for global couplng, such as through a coarse grd problem based on the coarse trangulaton f g. Accordngly, let R T H denote an nterpolaton map (say pecewse lnear nterpolaton) from the nodal values on V (vertces of subdomans) onto all the nodes on B. Then, R H can be vewed as the weghted restrcton map from B onto V. Note that the range of R T H here s B nstead of the whole doman. A varant M 2 of the precondtoner proposed by Bramble et al. (1986a) s a smple modcaton of M 1 : M 1 2 = mx =1 R T E S 1 E E R E + R T H A 1 H R H; (3:6) where A H s the coarse grd dscretzaton as n Secton 2. Wth the global communcaton of nformaton, the rate of convergence of the algorthm becomes logarthmc n H=h. Theorem 6 There exsts a constant C ndependent of H, h such that cond (M 1 2 S) C 1 + log 2 (H=h) :

22 82 T.F. Chan and T.P. Mathew In case the coecents a are constant n each subdoman, then C s also ndependent of a. Proof. See Bramble et al. (1986a), Wdlund (1988) and Dryja et al. (1993). As for the precondtoner M 1 to ecently mplement ths algorthm, t s necessary to replace the subblocks S E E by sutable precondtoners, such as those descrbed for the two subdoman case n Secton 3.1, see also Chan, Mathew and Shao (1992b) Two-dmensonal case: vertex space precondtoner M 3 for S The logarthmc growth (1 + log(h=h)) 2 n the condton number of the precedng precondtoner M 2 can be elmnated at addtonal cost, by modfyng the BPS algorthm to result n the vertex space precondtoner proposed by Smth (1990, 1992). The basc dea s to nclude addtonal overlap between the subblocks used n the BPS precondtoner M 2. Recall that the Schur complement S s not block dagonal n the permutaton [E 1 ; : : :; E m ; V ], snce adjacent edges are coupled, wth S E E j 6= 0 whenever edges E and E j are part of the boundary of the same subdoman. Ths couplng was gnored n the precedng two precondtoners, and resulted n the logarthmc growth factor n the condton number. By ntroducng overlappng subblocks, one can provde sucent approxmaton of ths couplng, resultng n optmal convergence bounds. Overlap n the decomposton of nterface B = fe 1 [ [ E m g [ V; can be obtaned by ntroducng vertex regons fv S 1 ; : : :; V S q g centred about each vertex n V (assume there are q subdoman vertces): B fe 1 [ [ E m g [ V [ fv S 1 [ V S q g: The vertex regons V S k are llustrated n Fgure 3, and are dened as the cross shaped regons centred at each subdoman vertex (x H k ; yh k ) contanng segments of length H of all the edges E that emanate from t. Such vertex spaces were used earler by Nepomnyaschkh (1984; 1986). Correspondng to ths overlappng cover of B, we denote the ndces of the nodes that le on E by E, the ndces of the vertces by V, and the ndces of the vertex regon V S by V S. Thus E 1 [ [ E m [ V [ V S 1 [ V S q form an overlappng collecton of ndces of all unknowns on B. As wth the restrcton and extenson maps for the BPS, we let R V S denote the restrcton of full vectors to subvectors correspondng to the ndces n V S. Its

23 Doman decomposton survey 83 transpose R T V S denotes the extenson by zero of subvectors wth ndces V S to full vectors. The prncpal submatrx of S correspondng to the ndces V S wll be denoted S V S = R V S SR T V S. The vertex space precondtoner M 3 s an addtve Schwarz precondtoner dened on ths overlappng partton: M 1 3 = mx =1 R T E S 1 E E R E + R T HA 1 H R H + A T V S ;V S qx =1 A V S ;V S R T V S S 1 V S R V S : (3:7) In general, the matrces S V S are dense and expensve to compute. However, sparse approxmatons can be computed ecently usng the probng technque or modcatons of Dryja's nterface precondtoner by Chan et al. (1992b). Alternately, usng the followng approxmaton: SV 1 S z V S 0 " # 1 A V S A V S ;V S 0 I ; z V S the acton of S 1 V S can be approxmated by solvng a Drchlet problem on a doman V S of dameter 2H whch contans V S and whch s parttoned nto a small number (four for rectangular regons) subregons by the nterface V S. The convergence rate of the vertex space precondtoned system s optmal n H and h (but may depend on varatons n the coecents). Theorem 7 that There exsts a constant C 0 ndependent of H, h and such cond (M 1 3 S) C 0(1 + 1 ); where C 0 may depend on the varatons n a. There also exsts a constant C 1 ndependent of H, h, and the jumps n a (provded a s constant on each subdoman ) but can depend on such that Proof. cond (M 1 3 S) C 1(1 + log(h=h)): See Smth (1992), Dryja et al. (1993) and also Secton 4. Thus, n the presence of large jumps n the coecent a, the condton number bounds for the vertex space algorthm may deterorate to (1 + log(h=h)), whch s the same growth as for the BPS precondtoner Two-dmensonal case: Neumann{Neumann precondtoner M 4 for S The Neumann{Neumann precondtoner for S n the case of many subdomans s a natural extenson of the Neumann{Neumann algorthm for the case of two subregons, descrbed n Secton 3.1. Ths precondtoner was orgnally proposed by Bourgat et al. (1989), and extended by De Roeck (1989), De Roeck and Le Tallec (1991), Le Tallec, De Roeck and Vdrascu

24 84 T.F. Chan and T.P. Mathew (1991), Dryja and Wdlund (1990; 1993a,b), Mandel (1992) and Mandel and Brezna (1992). There are several versons of the Neumann{Neumann algorthm, wth the derences arsng n the choce of a mechansm for global communcaton of nformaton. We follow here a verson due to Mandel and Brezna (1992), referred to as the balancng doman decomposton precondtoner. Neumann{Neumann refers to the process of solvng Neumann problems on each subdoman durng each precondtonng step. For each subdoman let denote the pontwse restrcton map (matrx) from nodes on B nto nodes \ B. Its transpose R denotes an extenson by zero of nodal values \ B to the rest of B. Correspondng to subdoman, we denote the stness matrx of the Neumann problem by " # () A II A () IB where A () II nteror of, A () A () A ()T IB A () BB s a prncpal submatrx of A correspondng to the nodes n the IB s a submatrx of A correspondng to the couplng between nodes n the nteror of and the nodes on the nterface B restrcted and A () BB corresponds to the couplng between the nodes wth contrbutons from (n the nte element case, A () s obtaned by ntegratng the weak form on for all the bass functons correspondng to the nodes ). For each subdoman, we let S () denote the Schur complement wth respect to the nodes \ B of the local stness matrx A () : S () = A () BB A()T A () IB : (3:8) The natural extenson of the two subdoman Neumann{Neumann precondtoner s smply M ~ 4 : ~M 1 4 = px =1 1 IB A() II ; BB R D S () 1 D ; (3:9) where D s a dagonal weghtng matrx. Note that (S () ) 1 v can be computed by a Neumann solve wth v as Neumann data (see Secton 3.1). Ths precondtoner s hghly parallelzable, but t has two potental problems: The matrx S () s sngular for nteror subdomans snce t corresponds to a Neumann problem on. Accordngly, a compatblty condton must be satsed, and addtonally, the soluton of the sngular system wll not be unque. There s no mechansm for global communcaton of nformaton, and hence the condton number of the precondtoned system deterorates at least as H 2.

25 Doman decomposton survey 85 One way to rectfy these two defects s the balancng procedure of Mandel and Brezna (1992). The resdual s projected onto a subspace whch automatcally satses the compatblty condtons for each of the sngular systems (as many as p constrants). Addtonally, n a post processng step, a constant s added to the soluton of each local sngular system so that the resdual remans n the approprate subspace. Ths procedure also provdes a mechansm for global communcaton of nformaton. We omt the techncal detals, and refer the reader to Mandel and Brezna (1992). The sngularty of the local Neumann problems also arses n a related method by Farhat and Roux (1992) where the nterface compatblty condtons are enforced by a Lagrange multpler approach. The moded Neumann{Neumann precondtoner M 4 (wth balancng) satses: Theorem 8 There exsts a constant C ndependent of H and h and the jumps n the coecents a such that cond (M 1 4 S) C (1 + log(h=h))2 : Proof. See De Roeck and Le Tallec (1991), Mandel and Brezna (1992), Dryja and Wdlund (1993a). The Neumann{Neumann precondtoner has several attractve features: the subregons need not be trangular or rectangular; they can have general shapes; no explct computaton of the entres of S; the rate of convergence s logarthmc n H=h and nsenstve to large jumps n the coecents a. However, the Neumann{Neumann precondtoner requres twce as many subdoman solves per step as a multplcaton wth S Three-dmensonal case: vertex space precondtoner M 1 for S Constructng eectve precondtoners for the Schur complement matrx S s more complcated n three dmensons. These dcultes arse n part from the ncreased dmenson of the boundares of three-dmensonal regons, and s also, techncally, from a weaker Sobolev nequalty n three dmensons. As n the two-dmensonal case, we assume that s parttoned nto p nonoverlappng subregons wth nterface B: = 1 [ [ p [ B; where B = ([ p ) \ : For most of the three-dmensonal algorthms we wll descrbe, t wll be assumed that the f g consst of ether tetrahedrons or cubes and form a coarse trangulaton of havng mesh sze H. The of

26 86 T.F. Chan and T.P. Mathew each tetrahedron or cube can be further parttoned nto faces, edges and vertces. The faces F j = nteror j are assumed to be open two-dmensonal surfaces. The edges E k are one-dmensonal curves dened to be the ntersecton of the boundares of two faces: E k j ln excludng the endponts. Fnally, the vertces V are pont sets whch are the endponts of edges. As a prelude, we descrbe two precondtoners M 1a and M 1b related to the vertex space precondtoner M 1. Correspondng to the partton of B nto faces, edges and subdoman vertces, we permute the unknowns on B as x B = [x F ; x E ; x V ] T ; where F denote all the nodes on the faces, E corresponds to all the nodes on the edges E, whle V denotes all the subdoman vertces. Thus, the matrx S has the followng block form: S = 2 4 S F F S F E S F V S T F E S EE S EV S T F V S T EV S V V The rst precondtoner M 1a wll be a block dagonal approxmaton of the above block partton of S, wth the ncluson of a coarse grd model for global communcaton of nformaton, see Dryja et al. (1993). Accordngly, for each of the subregons of B, let R F, R Ek and R V denote the pontwse restrcton map from B onto the nodes on face F, edge E k and subdoman vertces V, respectvely. Ther transposes correspond to extensons by zero onto all other nodes on B. The prncpal submatrces of S correspondng to the nodes on F, E k and V wll be denoted by S F F, S Ek E k and S V V, respectvely. For the coarse grd problem, let R T H denote the nterpolaton map from the subdoman vertces V to all nodes on B. Then, ts transpose R H denotes a weghted restrcton map onto the subdoman vertces V. The coarse grd matrx s then gven by A H = R H AR T H. In terms of the restrcton and extenson maps gven above, M 1a s dened by M 1 1a = X R T F S 1 F F R F + X k 3 5 : R T E k S 1 E k E k R Ek + R T HA 1 H R H: We note that the couplng terms S F F j and S E E j between adjacent faces and edges have been dropped. For nte element and nte derence dscretzatons, the blocks S E E can be shown to be well condtoned (ndeed, for seven-pont nte derence approxmatons on three-dmensonal rectangular subdomans, S E E = A E E, snce boundary data on the edges do not nuence the soluton n the nteror of the regon). Consequently, S E E may be eectvely replaced by a sutably scaled multple of the dentty matrx M E E : S E E M E E = h E I E ; where E represents the average of the coecents a n the subdomans ad-