Numerische Mathematik

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1 Numer. Math. 89: (2001) Dgtal Object Identfer (DOI) /s Numersche Mathematk Algebrac theory of multplcatve Schwarz methods Mchele Benz 1,, Andreas Frommer 2, Renhard Nabben 3, Danel B.Szyld 4, 1 Department of Mathematcs and Computer Scence, Emory Unversty, Atlanta, GA 30322, USA; e-mal: benz@mathcs.emory.edu 2 Fachberech Mathematk, Unverstät Wuppertal, Wuppertal, Germany; e-mal: frommer@math.un-wuppertal.de 3 Fakultät für Mathematk, Unverstät Belefeld, Postfach , Belefeld, Germany; e-mal: nabben@mathematk.un-belefeld.de 4 Department of Mathematcs, Temple Unversty, Phladelpha, PA , USA; e-mal: szyld@math.temple.edu Receved February 21, 2000 / Revsed verson receved July 12, 2000 / Publshed onlne Aprl 5, 2001 c Sprnger-Verlag 2001 Summary. The convergence of multplcatve Schwarz-type methods for solvng lnear systems when the coeffcent matrx s ether a nonsngular M-matrx or a symmetrc postve defnte matrx s studed usng classcal and new results from the theory of splttngs. The effect on convergence of algorthmc parameters such as the number of subdomans, the amount of overlap, the result of nexact local solves and of coarse grd correctons (global coarse solves) s analyzed n an algebrac settng. Results on algebrac addtve Schwarz are also ncluded. Mathematcs Subject Classfcaton (1991): 65F10, 65F35, 65M55 1.Introducton We consder the soluton of large sparse lnear systems of the form (1) Ax = b by multplcatve or addtve Schwarz methods. Our am s to apply the theory of matrx splttngs to study the convergence of these classes of methods, Ths work was performed under the auspces of the U.S. Department of Energy through grant W-7405-ENG-36 wth Los Alamos Natonal Laboratory. Ths work was supported by the Natonal Scence Foundaton grant DMS Correspondence to: D.B. Szyld

2 606 M. Benz et al. usng propertes of the coeffcent matrx only. Specfcally, we analyze two cases: the case where the coeffcent matrx A s a nonsngular M-matrx, and when A s symmetrc postve defnte (s.p.d.). As we shall see, n several stuatons there s a nce common theory n the treatment of these two cases, usng the approprate splttngs for each case. The exceptons are Sects. 5 and 6 where for multplcatve Schwarz only the M-matrx case s studed. Whle several convergence results on Schwarz methods exst when the matrx A n (1) corresponds to the dscretzaton of a dfferental equaton (see, e.g., [9], [40], [44], and the extensve bblography theren), there s a need to analyze these methods n a purely algebrac settng. As we show, there are nstances where the tools developed here provde convergence analyss not avalable wth the usual Sobolev space theory. We beleve that the algebrac and analytcal ponts of vew complement each other. Furthermore, there are applcatons, such as electrcal power networks and Leontef models n economcs, where the matrx A does not come from a dfferental equaton (and t s an M-matrx); see, respectvely, [10] and [2]. Another case of nterest s when the problem arses from the dscretzaton of a dfferental equaton but no geometrc nformaton about the underlyng mesh s avalable to the solver. Addtonally, an algebrac approach s useful for the case of unstructured meshes [7]. There are several papers wth detaled abstract analyss (.e., ndependent of the partcular dfferental equatons n queston) of Schwarz methods, ncludng those of Xu [51] and Grebel and Oswald [23], where A-norm bounds are obtaned for the symmetrc postve defnte case; see also the ncely wrtten survey [25]. In other cases, e.g., n [32], the maxmum prncple s used to show convergence. In ths paper we concentrate on the case of algebrac multplcatve Schwarz, although we nclude new results on addtve Schwarz, and hybrd methods as well. We emphasze methods where overlap s used,.e., when the same varable s present n more than one local solver. The present work can be seen as a contnuaton of [20] where algebrac addtve Schwarz was consdered, and t complements the heurstc study [7]. Whle we do not provde condton number estmates for a precondtoned system, our convergence results pont out to the usefulness of the multlevel methods as soluton methods as well as precondtoners for a wder class of problems. In the s.p.d. case we are able to prove convergence wthout the usual assumptons; see Remark 4.11 below. In the nonsymmetrc case, we can prove convergence wthout any condton on the coarse grd correcton, and n fact convergence s shown wthout the need for a coarse grd correcton; see Remark 3.6.

3 Algebrac theory of multplcatve Schwarz methods 607 Gven an ntal approxmaton x 0 to the soluton of (1), the (one-level) multplcatve Schwarz method can be wrtten as the statonary teraton (2) x k+1 = Tx k + c, k =0, 1,..., where (3) T =(I P p )(I P p ) (I P 1 )= and c s a certan vector. Here P = R T (R AR T ) R A 1 (I P ) where R s a matrx of dmenson n n wth full row rank, 1 p.in the case of overlap we have n >n. Note that each P, and hence each =1 I P, s a projecton operator;.e., (I P ) 2 = I P. Each I P s sngular and has spectral radus equal to 1. Yet, as we wll see, the product T gven by (3) has spectral radus strctly less than 1 under sutable assumptons. In fact, for an approprate norm I P =1, =1,...,p(as s well known for A s.p.d.) but the product matrx T has norm less than 1. The matrx R corresponds to the restrcton operator from the whole space to a subdoman Ω (of dmenson n ) n the doman decomposton settng, and the matrx A = R AR T s the restrcton of A to that subdoman. A soluton usng A s called a local solve, and ths name carres to the purely algebrac case. Our approach conssts n determnng the unque splttng A = B C wth B nvertble and such that T = B C, and to study the propertes of that splttng; see Lemma 2.1 below. In ths way we can explot the rch theory of matrx splttngs and prove convergence under approprate condtons. In ths paper we emphasze the use of Schwarz methods as solvers rather than precondtoners. We note that when used as a precondtoner, partcularly n the case of symmetrc postve defnte problems and the conjugate gradent method, the multplcatve Schwarz method s usually symmetrzed; that s, the applcaton of the p projectons n (3) s followed by another sweep of projectons appled n the reverse order. Many of the results and technques of ths paper can be appled to the symmetrzed teratons. There are a number of papers dealng wth algebrac Schwarz methods, ncludng [7], [16], [18], [19], [42], [43], [46], [53]; see also [29]. In many of these, only specal cases, such as trdagonal, or block-trdagonal matrces, or matrces derved from a partcular model problem, are studed. Our contrbuton s to provde convergence results for multplcatve Schwarz methods (wth overlappng blocks) for general M-matrces and for s.p.d. =p

4 608M. Benz et al. matrces. We present our convergence bounds n terms of matrx norms as well as spectral rad, and use both of these to compare the convergence of dfferent versons. In partcular, we analyze the effect on convergence of algorthmc parameters such as the number of blocks (or subdomans) p, the amount of overlap, nexact local solves, and the effect of addng coarse grd correctons (both multplcatvely and addtvely). 2.Auxlary results The purpose of ths secton s to ntroduce some notaton and a few results that wll be used extensvely n the remander of the paper. A matrx B s nonnegatve (postve), denoted B O (B > O) f ts entres are nonnegatve (postve). We say that B C f B C O, and smlarly wth the strct nequalty. These defntons carry over to vectors. A matrx A s a nonsngular M-matrx f ts off-dagonal elements are nonpostve, and t s monotone,.e., A O. It follows that f A and B are nonsngular M-matrces and A B, then A B [2], [49]. By ρ(b) we denote the spectral radus of the matrx B. A matrx B s symmetrc postve defnte (s.p.d.), denoted B O, f t s symmetrc and f for all vectors u 0, u T Bu > 0, and postve semdefnte, denoted B O, f for all vectors u 0, u T Bu 0. Wesay that B C f B C O, and smlarly wth the strct nequalty. It follows that f A and B are s.p.d. and A B, then A B.IfA O one can defne the A-norm of a vector x as x A =(x T Ax) 1/2. Ths vector norm nduces a matrx norm n the usual way. We say that A = M N s a splttng f M s nonsngular. The splttng s regular f M O and N O; t s weak regular f M O and M N O; and t s nonnegatve f M O, M N O, and NM O [2], [49], [50]. The splttng s P -regular f M T +N s postve defnte [39]. Note that f A s symmetrc M T + N = M T + M A s also symmetrc. We say that a splttng s a strong P -regular splttng of A s.p.d., when N O. Ths mples that M O and that n partcular t s a P -regular splttng. The followng result, whch can be found, e.g., n [1], shows that gven an teraton matrx, there s a unque splttng for t. Lemma 2.1. Let A and T be square matrces such that A and I T are nonsngular. Then, there exsts a unque par of matrces B, C, such that B s nonsngular, T = B C and A = B C. The matrces are B = A(I T ) and C = B A = A((I T ) I). The followng characterzaton of P -regular splttngs wll be useful n our analyss; for a proof, see [20], or [52].

5 Algebrac theory of multplcatve Schwarz methods 609 Lemma 2.2. Let A be symmetrc postve defnte. Then A = M N s a P-regular splttng f and only f M N A < 1. In ths paper we assume that the rows of R are rows of the n n dentty matrx I, e.g., R = Ths restrcton operator s often called a Boolean gather operator, whle ts transpose R T s called a Boolean scatter operator. Formally, such a matrx R can be expressed as (4) R =[I O] π wth I the dentty on R n and π a permutaton matrx on R n. In ths case, t follows that A s an n n prncpal submatrx of A. In fact, we can wrte [ ] π Aπ T A K (5) =, L A where A s the prncpal submatrx of A complementary to A,.e. (6) A =[O I ] π A π T [O I ] T wth I the dentty on R n n. Recall that f A s an M-matrx, so are ts prncpal submatrces, and thus both A and A are M-matrces [2]. Smlarly f A s s.p.d., then, both A and A are s.p.d. For each = 1,...,p, we construct dagonal matrces E R n n assocated wth R from (4) as follows (7) E = R T R. These dagonal matrces have ones on the dagonal n every row where R T has nonzeros. We further assume that f S s the set of ndexes of the rows of the dentty that are rows of R, then (8) p S = S = {1, 2,...,n}. =1 In other words each varable s n at least one set S. Ths s equvalent to sayng that E I, wth equalty f and only f there s no overlap. Note =1 that n the case of overlappng blocks, we have here that each dagonal entry of E s greater than or equal to one, whch mples nonsngularty. Only =1 n the rows correspondng to overlap ths matrx has an entry dfferent from

6 610 M. Benz et al. one. In the case of overlap, the maxmum that these entres can attan s q, the measure of overlap defned below. We thus have that E qi. Let us defne a measure of overlap q of the decomposton (8) as the mnmal number of sets V k (k =1,...,q) such that =1 (9) q p V k = S = S = {1, 2,...,n}, k=1 =1 where each S s a subset of some V k, and f (10) S V k and S j V k for the same k, j, then S S j =. In other words, the measure of overlap s q = max { : j S }, j=1,...,n and obvously q =1mples that there s no overlap. Followng Hackbusch [24, Ch. 11], we defne the number of colors q of the decomposton (8) as the number of sets V k satsfyng (9), (10), and n addton, f r S, s S j, then the matrx entres a rs = a sr =0. It follows that q q, and often ths nequalty s strct. Furthermore, q depends only on the partton of the varables, whle q also on the graph of the matrx A. As we shall see, these two quanttes are used n the study of convergence of the addtve Schwarz method. The measure of overlap s relevant n the M-matrx case, whle the number of colors n the s.p.d. case. We llustrate the concepts of measure of overlap and number of colors wth two examples. Consder the matrx A 11 O O A 1,4 O O A 2,2 O A 2,4 A 2,5 A = O O A 3,3 O A 3,5 A 4,1 A 4,2 O A 4,4 O, O A 5,2 A 5,3 O A 5,5 where all dagonal blocks A,, =1,...,5 are 2 2 matrces. Now let S = {2 1, 2}, =1,...,5,.e. there s no overlap. Hence, we have for the measure of overlap q =1. We have only one set V 1 = 5 =1 S. The number of colors s q =2wth V 1 = S 1 S 2 S 3,V 2 = S 4 S 5.Ifwe take S 1 = {1, 2, 7, 8}, S 2 = {3, 4}, S 3 = {5, 6, 9, 10}, S 4 = {7, 8}, S 5 = {9, 10}

7 Algebrac theory of multplcatve Schwarz methods 611 we have q =2wth V 1 = S 1 S 2 S 3 and V 2 = S 4 S 5. The number of colors s q =3wth V 1 = S 1 S 3, V 2 = S 2 and V 3 = S 4 S 5. If A s a nonsngular M-matrx, for each =1,...,p, we construct a second set of matrces M R n n assocated wth R from (4) as follows [ ] M = π T A O (11) π OD, where (12) D = dag(a ) O has postve entres along the dagonal and thus s nvertble. Proposton 2.3. Let A be a nonsngular M-matrx. Let M be defned as n (11). Then the splttngs A = M N are regular (and thus weak regular and nonnegatve). Proof. Observe that M = π T [ A O O D ] π s nonnegatve. Thus, M s an M-matrx. Moreover N = M A s nonnegatve, snce t s a symmetrc permutaton of a matrx wth a 2 by 2 block structure, the off-dagonal blocks beng nonnegatve and the dagonal blocks beng ether zero, or nonnegatve wth a zero dagonal. Wth the defntons (7) and (11) we obtan the followng equalty whch we wll use throughout the paper (13) E M = R T A R,=1,...,p. We note that the matrx M defned n (11) s dfferent from the one used n [20], although we obtan the same characterzaton (13). All results n [20] hold verbatm for ths dfferent choce of M. In fact, we have a great deal of flexblty n choosng the matrces M, as long as the equalty (13) holds. We wll take advantage of ths flexblty n sectons 4 6 when analyzng the change n the convergence rate by varyng the degree of overlap, the number of blocks (subdomans) and the level of nexactness of the local solves. For the analyss of the s.p.d. case, we choose a dfferent set of matrces M satsfyng (13), namely the choce made n [20]. We abuse the notaton, but n each case t s clear from the context whch matrx t s we are usng. Let A be s.p.d. For each =1,...,p, we construct matrces M R n n assocated wth R as follows [ ] M = π T A O (14) π OA. It follows that M s s.p.d., and that t satsfes the dentty (13).

8 612 M. Benz et al. Proposton 2.4. Let A be a symmetrc postve defnte matrx. Let M be defned as n (14). Then, the splttngs A = M N are P-regular. Proof. Snce A T = A and A T = A, we wrte [ ] M T + N = π T A K π L A. The followng dentty shows that ths matrx s s.p.d., and thus we have a P -regular splttng: [ ][ ][ ] [ ] π T I A K I π I L A I = π T A K π L A = A. Gven a postve vector w R n, denoted w>0, the weghted max-norm s defned for any y R n as y w = max 1 y j ; see, e.g., [26], [41]. j=1,,n w j As usual the matrx norm s defned as T w = Tx w, and can be obtaned as (see, e.g., [41]) (15) T w = max ( T w) w. sup x w=1 We pont out that for T O, Tw < γw mples T w <γ(γ >0) [41]. Weghted max norms play a fundamental role n the study of asynchronous methods (see [21], [45]), and are natural generalzatons of the usual max norm. Most of our estmates hold for all postve vectors w of the form w = A e, where e s any postve vector,.e., for any postve vector w such that Aw s postve. In partcular ths would hold for w = A e and e =(1,...,1) T,.e., wth w beng the row sums of A. Recall that for A a nonsngular M-matrx, A O, and that snce A s nonsngular, no row of t can be a zero row. Ths guarantees that w = A e>0. The same logc s used to conclude that M e>0for any monotone matrx M, and ths s also used n our proofs. In ths paper we wll use several comparson theorems. The frst relates the weghted max norms of the teraton matrces and can be found n [20], [37]. Theorem 2.5. Let A O, and let A = M N = M N be two weak regular splttngs of A wth (16) M M. Let w>0be such that w = A e for some e>0. Then, (17) M N w M N w < 1. If the nequalty (16) s strct, then the frst nequalty n (17) s also strct.

9 Algebrac theory of multplcatve Schwarz methods 613 We pont out that wth the same hypotheses of Theorem 2.5, an nequalty of the form (17) does not necessarly hold for the spectral rad; see a counterexample n [15]. The followng three lemmas are helpful n our comparsons of spectral rad. The frst one s well known, and can be found, e.g., n [31]. Lemma 2.6. Assume that T R n n s nonnegatve and that for some α 0 and for some nonzero vector x 0, we have Tx αx. Then ρ(t ) α. The nequalty s strct f Tx>αx. Lemma 2.7. Assume that A = M N = M N are two splttngs of A, that M N O, and that M N has an egenvector x 0 wth egenvalue ρ(m N) such that Ax 0. If (16) holds, then ρ(m N) ρ( M N). Proof. We have O M N = I M A. Therefore (I M A)x (I M A)x = ρ(m N)x, and the asserton follows from Lemma 2.6. Lemma 2.8. Let A be monotone. Let A = M N be a splttng such that M O and NM O (sometmes called weak nonnegatve of the second type). Then ρ(m N) < 1 and there exsts a non-zero vector x 0 such that M Nx = ρ(m N)x and Ax 0. Proof. Note frst that A T = M T N T s a weak regular splttng of A T wth A T O, and thus convergent [2], [49]. Therefore, ρ((m T ) N T )= ρ((nm ) T )=ρ(nm ) < 1. Moreover, snce the spectrum of NM s equal to that of M N we know that ρ(nm )=ρ(m N) s an egenvalue of M N. Let x 0be an egenvector of M N whch s scaled n such a way that not all ts components are negatve. We now frst prove the lemma by assumng not only NM O but NM >O. Then we have (denotng ρ = ρ(m N)) that M Nx = ρx and therefore (18) Nx =(NM )(Mx)=ρMx. By the Perron-Frobenus theorem (see, e.g., [2], [49]), the postve matrx NM has a postve egenvector y belongng to the egenvalue ρ and, up to scalng, ths egenvector s unque. Snce Mx 0we therefore have Mx = αy for some α 0, and thus x = αm y. But M y>0, and snce not all components of x are negatve we see that α>0 and therefore x>0 as well as Mx > 0. From (18) we have that and snce ρ<1 ths proves Ax > 0. Ax = Mx Nx =(1 ρ)mx

10 614 M. Benz et al. To complete the proof, assume now that NM O. Let E R n n be the matrx wth all entres 1 and take γ>0suffcently small such that the seres (γem ) ν converges. In ths case we have, snce EM >O, ν=0 O< (γem ) ν =(I γem ) ν=0 as well as O<B:= (I γem ) M. For all postve ε smaller than ε 0 = ρ(bma ) we consder the splttngs Then and A ε A ε = M (N + εbm). (N + εbm)m = NM + εb > O = A (I εbma ) = A (εbma ) ν O. By what we have already shown there exst postve vectors x ε such that M (N+εBM)x ε = ρ(m (N+εBM))x ε and A ε x ε > 0. We normalze these vectors to have norm 1 and put ε k = 1 k ε 0. Then the sequence x εk admts a convergent subsequence wth lmt x 0, x 0. By contnuty, ths x satsfes M Nx = ρx as well as Ax 0. The followng theorem of Woźnck [50] s now a drect consequence of Lemmas 2.7 and 2.8. Theorem 2.9. Let A O. Assume that A = M N = M N are two nonnegatve splttngs wth M M. Then, (19) ν=0 ρ(m N) ρ( M N) < 1. The nequalty (19) s strct f A >Oand M > M. The followng counterpart of Theorem 2.9 n the s.p.d. case s from [35]. Theorem Let A O. Assume that A = M N = M N are two (strong P -regular) splttngs wth O N N. Then, (19) holds. The frst nequalty (19) s strct f O N N. We conclude ths secton wth a new comparson theorem, whch s the counterpart to Theorem 2.5 usng A-norms, where A s s.p.d. We frst prove an ntermedate result.

11 Algebrac theory of multplcatve Schwarz methods 615 Lemma Let A O, and let A = M N be a splttng of A such that M s symmetrc. Then ρ(m N)= M N A. Proof. It follows from the followng denttes: M N A = I M A A = I A 1/2 M A 1/2 2 = ρ(i A 1/2 M A 1/2 ) = ρ(i M A) = ρ(m N). The followng theorem follows now drectly from Lemma 2.11 and Theorem Theorem Let A O, and let A = M N = M N be two (strong P -regular) splttngs of A wth (20) O N N. Then, (21) M N A M N A < 1. If the second nequalty n (20) s strct, then, the frst nequalty n (21) s also strct. The hypothess (20) cannot be weakened,.e., we need to assume that the matrces N 1 and N 2 are postve semdefnte matrces. Examples n [35] show that Theorems 2.10 and 2.12 are not true f one only assumes P -regular splttngs. 3.Convergence of the one-level method In ths secton we prove convergence of the one-level scheme (2) under the assumpton that the rows of R are rows of the n n dentty matrx I,.e., that R has the form (4). Recall the defnton of the sets S n (8). In general, the S are not dsjont. When they are, we have the multplcatve Schwarz method wthout overlap. The followng mportant lemma covers both cases (overlappng and non-overlappng). Lemma 3.1. Let A be monotone, and let a collecton of p trples (E,M,N ) be gven such that O E I, E I, and A = M N s a weak =1 regular splttng for 1 p. Let (22) T =(I E p Mp A)(I E p Mp A) (I E 1M1 A). Then for any vector w = A e>0wth e>0, ρ(t ) T w < 1. Furthermore, (23) I E M A w =1, =1,...,p.

12 616 M. Benz et al. Proof. In order to show that T w < 1, where T w denotes the maxmum weghted norm of T wth respect to a certan vector w>0, we show that T O and Tw < w. Clearly T O because for =1,...,p, I E M A = I E + E (I M A) (24) = I E + E M N O and M N O snce the splttngs are weak regular. Next, we show that Tw < w wth w = A e where e>0. Tobegn wth, note that w 1 := (I E 1 M1 A)w = w E 1M1 e 0. Hence, 0 w 1 w, wth strct nequalty n the components correspondng to S 1. In other words, denotng wth (w 1 ) the th component of w 1,wehave (w 1 ) { = w f / S 1 ; <w f S 1. Now let w 2 := (I E 2 M 2 A)w 1, we clam that w 2 w, and that n the components correspondng to S 2, the nequalty s strct. Indeed, 0 (I E 2 M2 A)w 1 =(I E 2 M2 A)(w 1 w+w) (I E 2 M2 A)w. Now observe that { =(w1 ) (w 2 ) w f / S 2 ; <w f S 2, snce S 2 mples that (w 2 ) =[(I E 2 M2 A)(w 1 w)] +(w E 2 M2 e) <w. Smlarly, one can show that for all k p 1, { =(wk ) (w k+1 ) f / S k+1 ; <w f S k+1. p Because S = {1, 2,...,n}, we conclude that Tw < w. It follows that =1 T w < 1 and therefore ρ(t ) < 1. To complete the proof, observe that we have shown that for each =1,...,p, and each j =1,...n (25) (I E M Aw) j w j, and thus I E M A w 1. Ths upper bound s attaned snce we have shown the nequalty n (25) s actually an equalty for j/ S, cf. (15).

13 Algebrac theory of multplcatve Schwarz methods 617 Remark 3.2. Lemma 3.1 holds for any monotonc norm,.e., a norm for whch 0 v w mples v w. In fact, ths s also the case for many other results n ths paper. One of the exceptons s Theorem 4.7 n the next secton, where the weghted max norm cannot be easly replaced. Remark 3.3. The collecton of trples {(E,M,N )} p =1 can be thought of as a multplcatve multsplttng of A, n analogy wth the standard (addtve) multsplttng of a matrx n the sense of O Leary and Whte [38]; see also [6] and the extensve bblography theren, and [36] for further extensons. Remark 3.4. In the specal case where E = I for all =1, 2,...,p,we obtan an extenson to the case of p splttngs of Theorem 3.2 n [1]; see also the remarks at the end of Sect. 3 n [1]. Lemma 3.1, together wth the characterzaton (13) and Lemma 2.1, s the fundamental tool for provng the convergence of the multplcatve Schwarz method for nonsngular M-matrces. Theorem 3.5. Let A be a nonsngular M-matrx. Then the multplcatve Schwarz teraton (2) converges to the soluton of Ax = b for any choce of the ntal guess x 0. In fact, for any w = A e>0 wth e>0, we have ρ(t ) T w < 1. Furthermore, there exsts a unque splttng A = B C such that T = B C, and ths splttng s nonnegatve. Proof. Let E be as n (7) and M as n (11). Observe that O E I, 1 p. The key to the proof s the characterzaton (13), from whch we have (26) I P = I E M A, 1 p. Moreover, by Proposton 2.3, the splttngs A = M N (wth N = M A) are regular. Hence, by Lemma 3.1, ρ(t ) T w < 1 for any w = A e>0wth e>0, and the teraton (2) converges for any ntal vector x 0. Furthermore, by Lemma 2.1, there exsts a unque splttng A = B C such that T = B C. To prove that the splttng s nonnegatve we begn by showng that B =(I T )A s nonnegatve or, equvalently, that B z 0 for all z 0. Lettng v = A z 0, all we need to show s that (I T )v 0, ortv v. Ths s proved n the same way as Lemma 3.1. Hence, the unque splttng A = B C s weak regular. To show that t s nonnegatve we need to show that T = I AB s also nonnegatve. To see ths, note that T =(I P p )(I P p ) (I P 1 ), where P = AE M = AR T A R, n vew of the representaton (13). To complete the proof we show that each factor I P s nonnegatve. In fact, (27) I P T = I R T A T R A T = I E M T A T O, just as n (24).

14 618M. Benz et al. Remark 3.6. In the analyss of multplcatve Schwarz for nonsymmetrc problems usng analytcal tools, convergence s only obtaned assumng the addton of a (multplcatve) coarse grd correcton, and furthermore that the coarse grd be fne enough; see, e.g., [8], [44, Sect. 5.4]. As can be observed, n the M-matrx case, our Theorem 3.5 (as well as Theorem 4.5 wth nexact solves) provdes convergence wthout a coarse grd correcton. In Sect. 7 we show convergence of the multplcatve Schwarz method wth a coarse grd correcton (both addtve and multplcatve) wthout any restrcton on how fne t s. We turn now to the counterpart to the convergence Theorem 3.5 n the s.p.d. case. To that end, we frst prove the followng lemma. Lemma 3.7. Let A be a symmetrc postve defnte matrx. Let x, y R n, such that (28) y =(I E M A)x, where E s defned n (7) and M n (14). Then the followng dentty holds: (29) y 2 A x 2 A = (y x) T E AE (y x) 0. Furthermore, (30) I E M A A =1, =1,...,p. Proof. Consder x = π T (xt 1,xT 2 )T and y = π T (yt 1,yT 2 )T, wth x 1,y 1 R n. Further, from (7) and (4) we have that [ E = π T I O (31) OO ] π. Consder now (28), whence we mmedately have that (32) y 2 = x 2, and usng (14) and (5), we also get (33) A y 1 = A 12 x 2, where here we use the notaton A 12 = K, and smlarly A 21 = L = A T 12. Usng these denttes we wrte y T Ay x T Ax =(y T 1,y T 2 )π Aπ T (y T 1,y T 2 ) T (x T 1,x T 2 )π Aπ T (x T 1,x T 2 ) T = y T 1 A y 1 + y T 2 A 21 y 1 + y T 1 A 12 y 2 x T 1 A x 1 x T 2 A 21 x 1 x T 1 A 12 x 2 = x T 2 A 21 (y 1 x 1 )+(y T 1 x T 1 )A 12 x 2 + y T 1 A y 1 x T 1 A x 1 = y T 1 A (y 1 x 1 ) (y T 1 x T 1 )A y 1 + y T 1 A y 1 x T 1 A x 1 = (y T 1 x T 1 )A (y 1 x 1 )= (y x) T E AE (y x),

15 Algebrac theory of multplcatve Schwarz methods 619 where the last equalty follows from the dentty [ ] E AE = π T A O π OO. Snce A s s.p.d., E AE s semdefnte, and the rght hand sde of (29) s nonpostve. Ths mples that I E M A A = I G A A 1, wth G = R T (R AR T ) R. To see that ths upper bound on the norm s attaned we wrte (I G A)x 2 A = x T Ax x T AG Ax. Snce G s semdefnte, let y be such that y T G y =0, e.g., y havng zeros n the entres correspondng to the nonzero columns of R as n (4). Then, for x = A y we have that (I G A)x 2 A = x 2 A. We note that the result (30) s well known; see, e.g., [24], [3]. Here we have shown a proof n terms of E and M smply for completeness, and to emphasze the smlarty wth (23). Theorem 3.8. Let A be a symmetrc postve defnte matrx. Then the multplcatve Schwarz teraton (2) converges to the soluton of Ax = b for any choce of the ntal guess x 0. In fact, we have ρ(t ) T A < 1. Furthermore, there exsts a unque splttng A = B C such that T = B C, and ths splttng s P-regular. Proof. As n the proof of Theorem 3.5 we have the relatons (26) followng as a consequence of the equaltes (13). Startng wth x (1) 0let x (+1) = (I P )x (). Thus x (p+1) = Tx (1). Usng (29) repeatedly, and cancelng terms, we obtan (34) Tx (1) 2 A x (1) 2 A = (x (+1) x () ) T E AE (x (+1) x () ). =1 Snce E AE s postve semdefnte t follows that the rght hand sde of (34) s nonpostve. However, the rght hand sde s zero f and only f E (x (+1) x () )=0 for all, =1,...,p. The other n n components of x (+1) x () are also zero usng the same argument as n Lemma 3.7 to obtan (32). But ths mples x (p+1) = x (+1) = x () = x (1), =1,...,p. Thus x (1) must be a common fxed pont of (I P ) for all =1,...,p. However, the fxed ponts of the projectons (I P ) are just the vectors z R n wth E z =0. Snce E I there s no such common nonzero fxed pont. Hence the rght hand sde of (34) must =1

16 620 M. Benz et al. be negatve, and we obtan ρ(t ) T A < 1. Furthermore, by Lemma 2.1, there exsts a unque splttng A = B C such that T = B C. Wth Lemma 2.2 we obtan that ths nduced splttng s P -regular. Remark 3.9. In Lemma 3.7 and n Theorem 3.8t was not requred that the matrx (14) defne a P -regular splttng. Nevertheless, the product of the operators (26) produces a matrx wth an nduced splttng whch s P - regular. In fact, we have that n the (unsymmetrzed) multplcatve Schwarz method, B T + C s symmetrc and postve defnte. The convergence result of Theorem 3.8s not new; see, e.g., [9], [24, Ch. 11], [40], [44], [51]. Here we have gven a dfferent proof, as a counterpart to our new Theorem Inexact solves In ths secton we study the effect of varyng how exactly (or nexactly) the local problems are solved. We begn wth some results for algebrac addtve Schwarz. The addtve Schwarz method for the soluton of (1) s of the form (2), where (35) T = T = I =1 R T A R A, where 0 < 1s a dampng parameter; see [9], [11], [12], [13], [23], [24, Ch. 11], [44]. We emphasze that convergence of the damped addtve Schwarz method s only guaranteed for 1/q n the M-matrx case and for <1/ q n the s.p.d. case [20], [24, Ch. 11]. In fact, smple examples show that ths method may not be convergent for =1. Very often n practce, nstead of solvng the local problems A y = z exactly, such lnear systems are approxmated by à z where à s an approxmaton of A ; see, e.g., [5], and the above mentoned references. By replacng A wth à n (35) one obtans the damped addtve Schwarz teraton wth nexact local solves, and ts teraton matrx s then (36) T = I =1 R T à R A. In the M-matrx case we assume, as n [20], that the nexact solves correspond to monotone matrces and satsfy (37) à A. Notce that ths s equvalent to the condton that the splttngs (38) A = à (à A ) be regular splttngs.

17 Algebrac theory of multplcatve Schwarz methods 621 In the s.p.d. case we assume, as s generally done (see, e.g., [24, Ch. 11]), that the nexact solves correspond to s.p.d. matrces and satsfy (39) à A. Ths assumpton mples that (40) A = à (à A ) are P -regular splttngs. Condtons (37) and (39) are easly satsfed. Ths s the case, e.g., f à has a subset of the nonzeros of A (ncludng the dagonal). Ths last case ncludes many standard splttngs such as the dagonal, trdagonal, or trangular part, as well as block versons of them. The other notable example s ncomplete factorzatons à = L U where the nonzeros of the factors are n the locatons of the nonzeros of A, and n partcular ILU(0) [33]. In these cases, the nequalty (37) holds, or equvalently, we have (weak) regular splttngs [33], [48]. For examples of splttngs for whch the nequalty (39) holds see [35]. Another stuaton worth mentonng where (39) holds s when A s semdefnte and the nexact solver s defnte. Ths process s usually called regularzaton; see, e.g., [14], [30]. In [20] t s shown that the damped addtve Schwarz teratons wth nexact local solves converge n the M-matrx case under the condton (37) and 1/q. Furthermore, t s shown that the nduced splttngs correspondng to (35) and (36) A = M N = M Ñ are weak regular. Here we show, under the same condtons, that the convergence rate s slower than n the exact case, and that the more nexact the local solves are, the slower the convergence. Furthermore, we show that the splttngs nduced by (35) and (36) are actually nonnegatve, whch allows us to compare spectral rad. Theorem 4.1. Let A be a nonsngular M-matrx. Let à and Ā be nexact solves of A satsfyng à Ā A. Let the dampng factor 1/q, whch mples that the damped addtve Schwarz method s convergent. Then, T w T w T w, where w>0 s such that Aw > 0, and T s obtaned by replacng à by Ā n (36), =1,...,p. Moreover, ρ(t ) ρ( T ) ρ( T ), and the splttngs nduced by these teraton matrces are nonnegatve. Proof. Observe that (41) (42) M M = = =1 =1 R T A R T à R = R = =1 =1 E M O, E M O,

18 622 M. Benz et al. where (43) M = π T [ Ã O OD Snce (37) mples A (44) M ] π, and thus M = π T Ã,wehave M M, for =1,...,p, [ Ã O O D ] π. and consequently M. It was shown n [20] that the unque splttng A = M N nduced by T s weak regular. The same s true of the splttng A = M Ñ. It s not dffcult to show that these are actually nonnegatve splttngs. Consder the splttng nduced by T. All we need to show s that the matrx ˆT = I AR T A R =1 s nonnegatve. Takng the transpose of ths matrx and reasonng as n the proof of Theorem 3.4 n [20] t follows that ˆT O, hence I AM = N M O and the nduced splttng s nonnegatve. Thus, usng Theorem 2.5, we have that f w>0s such that Aw > 0, T w T w. Also, usng Theorem 2.9, we have that ρ(t ) ρ( T ). The other nequaltes follow n the same manner. When A s s.p.d. and the nexact solves satsfy (39), convergence holds f <1/ q, as shown, e.g., n [24, Ch. 11]. Furthermore, the nduced splttng defned by M s P -regular; see [20]. Here we show that under the same hypotheses the convergence usng the nexact solves s slower as measured usng ether the spectral rad or the A-norm (these two quanttes beng equal n vew of Lemma 2.11). Furthermore, the more nexact the local solves are, the slower the convergence. We wll use he followng result for s.p.d. matrces whch can be found, e.g., n [24]. Lemma 4.2. Let A be a symmetrc postve defnte matrx, and A = R AR T, R a restrcton operator, so that A s a prncpal submatrx of A. Then R T A R A. Ths result s used n [24, Lemma ()], and n other references (e.g., [44]) to obtan drectly the bound (45) A pm, and further mprove t to (46) A qm, where M s M for the value =1.

19 Algebrac theory of multplcatve Schwarz methods 623 Theorem 4.3. Let A be a symmetrc postve defnte matrx. Let à and Ā be nexact solves of A satsfyng à Ā A. Let the dampng factor <1/ q, whch mples that the damped addtve Schwarz method s convergent. Then, T A T A T A, where T s obtaned by replacng à by Ā n (36), =1,...,p. Moreover, ρ(t ) ρ( T ) ρ( T ), and the splttngs nduced by these teraton matrces are strong P -regular. Proof. Consder the matrces (41) and (42) whch are symmetrc postve defnte usng M as n (14) and (47) M = π T [ à O OA ] π. M Snce (39) mples A Ã, we have that M O. Ths mples M M and N Ñ. The theorem wll follow from Theorems 2.10 and 2.12 once we establsh N O,.e., that the splttngs are strong P -regular. To that end we use (46), and snce < 1/ q, wehaven = M A = 1 M A O. Remark 4.4. For smplcty, n Theorems 4.1 and 4.3, we assumed that the nexact versons use the same dampng parameter. It s evdent from the proofs that f the dampng parameter for the nexact verson s smaller, say <, the same conclusons hold. We proceed now wth smlar results for multplcatve Schwarz wth nexact solves. In ths case, the teraton matrx s (48) T =(I E p M p A)(I E p M p A) (I E 1 M 1 A), cf. (22). We frst prove convergence n the M-matrx case, and proceed wth comparsons varyng the amount of nexactness of the local solves. Theorem 4.5. Let A be a nonsngular M-matrx. Then the multplcatve Schwarz teraton wth teraton matrx (48) and wth nexact solves satsfyng (37) converges to the soluton of Ax = b for any choce of the ntal guess x 0. In fact, for any w = A e>0wth e>0, we have ρ( T ) T w < 1. Furthermore, there exsts a unque splttng A = B C such that T = B C, and ths splttng s nonnegatve. Proof. The proof proceeds n the same manner as that of Theorem 3.5. All we need to show s that each splttng A = M Ñ, wth M as n (43) s regular. Snce à s monotone, t follows from (43) that M O. Now, Ñ = M A and [ ] π Ñ π T à A = K, L D A whch, n vew of (37), (12), and the fact that A s an M-matrx, s nonnegatve.

20 624 M. Benz et al. Remark 4.6. Theorem 4.5 holds wth weaker hypotheses, namely, that the splttngs A = à (à A ) are weak regular splttngs,.e., that à (à A ) O, cf. (37). Ths s the same assumpton used n [20], and t mples that the splttngs A = M Ñ are weak regular. Theorem 4.7. Let A be a nonsngular M-matrx. Let à and Ā be nexact solves of A satsfyng à Ā A. Then, ρ(t ) ρ( T ) ρ( T ), and for any w>0 such that Aw > 0 we have T w T w T w < 1, where T s obtaned by replacng à by Ā n (48), =1,...,p. Proof. We start by establshng the nequaltes for the spectral rad. We confne ourselves to show ρ(t ) ρ( T ); the nequaltes for ρ( T ) are proved n the same way. By Theorem 3.5 both teraton matrces, T and T, arse from nonnegatve splttngs of A. Let x 0, x 0be an egenvector of T wth egenvalue ρ(t ). We wll show that (49) Tx Tx = ρ(t )x so that by Lemma 2.6 we get the desred result ρ( T ) ρ(t ). Let x 0 = x 0 = x and defne x := (I E M A)x and x := (I E M A) x,= 1,...,p. Thus, x p = Tx and x p = Tx. To establsh (49) we proceed by nducton and show that (50) and (51) Ax 0, =1,...,p 1, 0 x x,=1,...,p. We then have (49) snce x p = Txand x p = Tx; see (22) and (48). For =0, (51) holds by assumpton, whle relaton (50) s true by Lemma 2.8(here t s crucal that the nduced splttngs are nonnegatve). Assume now that (50) and (51) are both true for some. To obtan (50) for +1, observe that Ax +1 = A(I E M (27) we have I AE M A)x =(I AE M )Ax.By O and Ax 0 by the nducton hypothess, and thus (50) holds for +1. To prove that (51) holds for +1, we use (44), (50), and the nducton hypothess to obtan x +1 =(I E M A)x (I E M A)x (I E M A) x = x +1. To establsh the nequaltes for the weghted max norms, one proceeds n precsely the same manner as before (usng w nstead of x) to show Tw Tw. Snce both matrces are nonnegatve, we get T w T w.

21 Algebrac theory of multplcatve Schwarz methods 625 Remark 4.8. The purpose of usng nexact local solves à n leu of A s to obtan convergence n less computatonal tme. Theorems 4.1 and 4.5 ndcate that, as to be expected, asymptotcally the nexact methods have slower convergence rate. Nevertheless, they converge n less computatonal tme f the savng from the nexact local solve s suffcently large to offset the loss n convergence rate. Ths s often the case n practce. We present now the counterpart to the convergence Theorem 4.5 for the s.p.d. case. Consder nexact solves à so that (40) holds. Note that we do not requre (39) to hold here. Frst we present a result smlar to Lemma 3.7, cf. [34]. Lemma 4.9. Let A be a symmetrc postve defnte matrx. Let x, y R n such that y =(I E M A)x where M s defned n (47) wth à satsfyng (40). Then the followng dentty holds: (52) y 2 A x 2 A = (y x) T E ( M T + M A)E (y x) 0. Furthermore, I E M A A =1, =1,...,p. Proof. The proof proceeds as that of Lemma 3.7. We have that (32) holds, but nstead of (33) we have Ãy 1 =(à A )x 1 A 12 x 2. We then obtan y T Ay x T Ax = x T 2 A 21 (y 1 x 1 )+(y1 T x T 1 )A 12 x 2 + y1 T A y 1 x T 1 A x 1 =(x T 1 (à A ) T y1 T ÃT )(y 1 x 1 )+ (y1 T x T 1 )((à A )x 1 Ãy 1 )+y1 T A y 1 x T 1 A x 1 =( x T 1 A (y1 T x T 1 )ÃT )(y 1 x 1 )+ (y1 T x T 1 )( A x 1 Ã(y 1 x 1 )) + y1 T A y 1 x T 1 A x 1 = (y1 T x T 1 )(à + ÃT A )(y 1 x 1 ) = (y x) T E ( M T + M A)E (y x) The rest of the proof s almost dentcal to that of Lemma 3.7. The followng theorem establshes the convergence of multplcatve Schwarz wth nexact solves n the s.p.d. case, and ts proof s almost dentcal to that of Theorem 3.8. Theorem Let A be a symmetrc postve defnte matrx. Then the multplcatve Schwarz teraton wth teraton matrx (48) wth M defned n (47) and wth nexact solves satsfyng (40) converges to the soluton of Ax = b for any choce of the ntal guess x 0. In fact, we have ρ( T ) T A < 1. Furthermore, there exsts a unque splttng A = B C such that T = B C, and ths splttng s P-regular.

22 626 M. Benz et al. Remark Our convergence Theorem 4.10 s qute general snce the nexact solves à need not be symmetrc as s requred n the standard treatment of Schwarz methods, e.g., n [44]. We only requre that à T + à A O. The parallel between the results for M-matrces and those for s.p.d. matrces s not complete. We do not have at the moment a counterpart for Theorem Varyng the amount of overlap We study here how varyng the amount of overlap between subblocks (subdomans) nfluences the convergence rate of both addtve and multplcatve Schwarz. Let us consder two sets of subblocks (subdomans) of the matrx A, as defned by the sets (8), such that one has more overlap than the other,.e., let (53) Ŝ S, =1,...,p, p p wth Ŝ = S = S. We make the natural assumpton that the larger =1 =1 sets do not ntersect wth other sets from the same group of varables V k,.e., that the measure of overlap q does not change. Of course, each set Ŝ defnes an ˆn n matrx ˆR, where ˆn s the cardnalty of Ŝ, and the correspondng n n matrx Ê = ˆR T ˆR, as n (7). The relaton (53) mples that (54) I Ê E O. Smlarly, f ˆπ s such that ˆR =[I O]ˆπ, wth I the dentty n Rˆn,we denote by  the correspondng prncpal submatrx of A,.e.,  = ˆR A ˆR T =[I O] ˆπ A ˆπ T [I O] T, and, as n (11) defne (55) ˆM =ˆπ T [  O O ˆD ] ˆπ, where ˆD = dag(â ) O, and  s the (n ˆn ) (n ˆn ) complementary prncpal submatrx of A as n (6). As n (13), we have here also the fundamental dentty Ê ˆM = ˆR T  ˆR, =1,...,p.

23 Algebrac theory of multplcatve Schwarz methods 627 We want to compare ˆM wth M, although  and A are of dfferent sze. Wthout loss of generalty, we can assume that the permutatons π and ˆπ concde on the set S, and that the ndexes n S are the frst n elements n Ŝ. In fact, we can assume that ˆπ = π. Thus, A s a prncpal submatrx of Â, and ˆM has the same dagonal as M. Snce both  and ˆM are M-matrces, t follows that (56) ˆM M,=1,...,p. We consder frst the case of damped addtve Schwarz wth teraton matrx (35), and the teraton matrx correspondng to the larger overlap s (57) ˆT = I =1 ˆR T  ˆR A. Before we state our frst comparson result, let us menton that n general one expects the ncrease of overlap defned by (53) to be such that the groupngs of the sets s mantaned, and thus the measure of overlap, q, to be the same. Ths s s not a constrant; f we have a dfferent measure of overlap, and say, ˆq q, we only need to change our hypothess 1/q to 1/ max{q, ˆq}. Theorem 5.1. Let A be a nonsngular M-matrx. Consder two sets of subblocks of A defned by (53), and the two correspondng addtve Schwarz teratons (35) and (57). Let the dampng factor 1/q, whch mples that the addtve Schwarz methods are convergent. Then, ˆT w T w, where w>0 s such that Aw > 0. Also, ρ( ˆT ) ρ(t ). Proof. Because M and ˆM are both M-matrces, t follows from (56) that (58) ˆM and together wth (54) we have ˆM = =1 ˆR T  ˆR = M,=1,...,p, Ê ˆM =1 Ê ˆM E M. Ths mples that =1 E M = M O, ˆM where A = ˆM ˆN s the unque splttng such that ˆT = ˆN ; see Lemma 2.1. Snce the splttngs are nonnegatve (see Theorem 4.1), the conclusons follow from Theorem 2.5 and Theorem 2.9. Theorem 5.2. Let A be a symmetrc postve defnte matrx. Consder two sets of subblocks of A defned by (53), and the two correspondng addtve Schwarz teratons (35) and (57). Let the dampng factor 1/ q, whch mples that the addtve Schwarz methods are convergent. Then, ˆT A T A and ρ( ˆT ) ρ(t ).

24 628M. Benz et al. Proof. Let Q = E M = R T A R and ˆQ = Ê ˆM = ˆR T Â ˆR. Snce A s a prncpal submatrx of Â, by Lemma 4.2 we have that ˆQ Q. Therefore, ˆM = ˆQ =1 =1 Q = M O. As shown n the proof of Theorem 4.3, these splttngs are strong P -regular, and the theorem follows from Theorems 2.10 and We remark that the proof of Theorem 5.2 does not really use the new representaton (13), but t s of the same sprt as the proof of Theorem 5.1. Theorems 5.1 and 5.2 ndcate that the more overlap there s, the faster the convergence of the algebrac addtve Schwarz method. As a specal case, we have that overlap s better than no overlap. Ths s consstent wth the analyss for grd-based methods; see, e.g., [4], [44]. In a way smlar to that descrbed n Remark 4.8, the faster convergence rate brngs assocated an ncreased cost of the local solves, snce now they have matrces of larger dmenson and more nonzeros. In the cted references a small amount of overlap s recommended, and the ncrease n cost s usually offset by faster convergence. Remark 5.3. Results smlar to Theorem 5.1 were shown for (addtve) multsplttng methods n [17] and [28]; see also [22]. In these references, though, the weghtng matrces had to be the same for both sets of splttngs. Here we are able to prove ths more general result snce we do not requre that E = Ê = I, as n the multsplttng settng. Instead all we need s =1 =1 that these sums be nvertble. We consder now the algebrac multplcatve Schwarz teraton wth (22) and the correspondng one wth the larger overlap,.e., (59) ˆT =(I Êp ˆM p A)(I Êp ˆM pa) (I Ê1 ˆM 1 A). Convergence follows n the M-matrx case from Theorem 3.5. Theorem 5.4. Let A be a nonsngular M-matrx. Consder two sets of subblocks of A defned by (53), and the two correspondng multplcatve Schwarz teratons (22) and (59). Then, ρ( ˆT ) ρ(t ), and for any vector w>0 such that Aw > 0 we have ˆT w T w. Proof. The proof proceeds exactly as n the proof of Theorem 4.7 usng (58).

25 Algebrac theory of multplcatve Schwarz methods Varyng the number of blocks We address here the followng queston. If we partton a block nto smaller blocks, how s the convergence of the Schwarz method affected? In the M- matrx case, we show that for both addtve and multplcatve Schwarz the more subblocks (subdomans) the slower the convergence. In the s.p.d. case, ths s shown only for addtve Schwarz. In a lmtng case, f each block s a sngle varable, ths s slower. Ths result s consstent wth the classcal comparson theorem of Varga [49], whch for example shows that the pont Jacob (pont Gauss-Sedel) method s asymptotcally slower than block Jacob (block Gauss-Sedel). As n the stuatons descrbed n sectons 4 and 5, the slower convergence may be partally compensated by less expensve local solves, snce they are of smaller dmenson. Formally, consder each block of varables S parttoned nto k subblocks,.e., we have (60) S j S, j =1,...,k, k j=1 S j = S, and S j S k = f j k. Each set S j has assocated matrces R j and E j = R T j R j. Snce we have a partton, (61) E j E, j =1,...,k, and k j=1 E j = E,=1,...,p. We defne the matrces A j = R j AR T j, and M j correspondng to the set S j n the manner already famlar to the reader (see, e.g., (55)), so that E j M j = R T j A j R j,j=1,...,k,=1,...,p. Gven a fxed dampng parameter, the teraton matrx of the refned partton s then k (62) T = I E j M j A, =1 j=1 cf. (35), and the unque nduced splttng A = M N (whch s a weak regular splttng) s gven by M = k E j M j. =1 j=1 We note that due to the ncluson (60), the measure of overlap q cannot ncrease.

26 630 M. Benz et al. Theorem 6.1. Let A be a nonsngular M-matrx. Consder two sets of subblocks of A defned by (8) and (60), respectvely, and the two correspondng addtve Schwarz teratons (35) and (62). Let the dampng factor 1/q, whch mples that the addtve Schwarz methods are convergent. Then, T w T w, where w>0 s such that Aw > 0. Furthermore, ρ(t ) ρ( T ). Proof. In the same way that the ncluson (53) mples the nequalty (56) and n turn the nequalty (58), here the ncluson (60) mples that (63) M j Combnng (61) wth (63) we have that and thus, M 2.9. k j=1 M, j =1,...,k,=1,...,p. E j M j k j=1 E j M = E M M, whch mples the result, usng Theorems 2.5 and Theorem 6.2. Let A be a symmetrc postve defnte matrx. Consder two sets of subblocks of A defned by (8) and (60), respectvely, and the two correspondng addtve Schwarz teratons (35) and (62). Let k = max k, and let the dampng factors be 1/ q, and = /k 1/(k q). Ths mples that the addtve Schwarz methods are convergent. Then, T A T A and ρ(t ) ρ( T ). Proof. As n the proof of Theorem 5.2 we have, usng Lemma 4.2, that Q j = E j M j k Therefore, Q j k Q, and j=1 M = k Q j k =1 j=1 M Q = E M. =1 M Q = km, whch s equvalent to = (1/k) M. The theorem now follows usng Theorems 2.10 and 2.12, and the fact that these are strong P -regular splttngs, as shown n the proof of Theorem 4.3. We note that the fact that the bound for the dampng factor s lower than that for s consstent wth the fact that we ncrease the number of regons n the same proporton. Nevertheless, the result of Theorem 6.2 holds for

27 Algebrac theory of multplcatve Schwarz methods 631 the same dampng factor. Ths follows from the fact that due to (60), the number of colors does not ncrease. The proof has to be modfed usng the same arguments as n [24, Lemma ] to mprove the bound (45) to obtan (46). Next, we consder the case of multplcatve Schwarz. Agan, we can show that usng more subblocks of smaller sze results n slower asymptotc convergence rates. The teraton matrx for the multplcatve Schwarz method correspondng to the fner partton (more subblocks) s gven by (64) 1 T = =p 1 j=k (I P j ), where P j = E j M j A = R T j A j R j A. Theorem 6.3. Let A be a nonsngular M-matrx. Consder two sets of subblocks of A defned by (8) and (60), respectvely, and the two correspondng multplcatve Schwarz teratons (3) and (64). Then ρ(t ) ρ( T ), and T w T w for any vector w>0for whch Aw > 0. Proof. Snce each P = E M A = R T A R s a projecton we have I P =(I P ) 2 =...=(I P ) k. Ths allows us to represent T from (3) (or (22)) as a product wth the same number of factors k = k as n the representaton (64) for T, namely =1 (65) 1 T = (I P ) k. =p We par each of the k factors I P j = I E j M j A of T n (64) wth the correspondng factor I P = I E M A of T n (65). Ths par of factors correspond to the set of ndces S j and S satsfyng S j S.By (61) and (63) we have that E j M j E M. We can therefore proceed n exactly the same manner as n the proofs of Theorems 4.7 and 5.4 to establsh the desred results. 7.Two-level schemes In ths secton we assume that all local solves are exact; however, analogous results hold for the case of nexact solves, provded that the condtons spelled out n Sect. 4 are satsfed. Suppose a coarse grd correcton s

28 632 M. Benz et al. added (multplcatvely) to the multplcatve Schwarz teraton (2). Ths results n a statonary method wth an teraton matrx of the form (66) H =(I G 0 A)T where T s the teraton matrx of the multplcatve Schwarz method and G 0 = R0 T (R 0AR0 T ) R 0. We assume here that R 0 s formed by some rows of the (n n) dentty matrx I, so that R 0 AR0 T s a prncpal submatrx of A. Typcally, R 0 s defned n such a way that t has at least one row n common wth each of the R matrces that defne the multplcatve Schwarz teraton, 1 p. Thus, the number of rows n R 0 s no less than p, and should be much less than n. In partcular, the coarse grd correcton proposed n [47] and used, e.g., n [27], s of ths form. As before, assocated wth ths matrx R 0, we defne matrces E 0 and M 0 such that E 0 M0 A = G 0A, and O E 0 I. Note that f A s an M-matrx, A = M 0 (M 0 A) s a regular splttng, and f A s s.p.d., M 0 O. The (sngular) matrx I G 0 A defnes the global coarse solve, whch follows the multplcatve Schwarz sweep. We are nterested n comparng the convergence rate of the multplcatve Schwarz teraton wth and wthout the coarse grd correcton. Theorem 7.1. Let A be a nonsngular M-matrx. Let T and H be the teraton matrces defned n (2) and (66), respectvely. Then ρ(h) ρ(t ), and for any vector w = A e>0 wth e>0, H w T w. Furthermore, the splttng nduced by H s nonnegatve. Proof. It s clear from Theorem 3.5 that addng a coarse grd correcton to the multplcatve Schwarz teraton preserves convergence: ρ(h) < 1. Hence, there exsts a unque splttng A = F (F A) such that H = I F A and the splttng s nonnegatve by Theorem 3.5. Furthermore, (67) F = B + G 0 (I AB ) B O, where A = B (B A) s the (unque) nonnegatve splttng nduced by T. By vrtue of (67) and Theorem 2.9 we conclude that ρ(h) ρ(t ), and usng Theorem 2.5, H w T w. Theorem 7.2. Let A be a symmetrc postve defnte matrx. Let T and H be the teraton matrces defned n (2) and (66), respectvely. Then ρ(h) H A T A < 1. Furthermore, the splttng nduced by H s P -regular. Proof. From Theorem 3.8, we have T A < 1, and from Lemma 3.7 we have that I G 0 A A =1. Hence H A = (I G 0 A)T A I G 0 A A T A = T A < 1. The nduced splttng s P -regular by Lemma 2.2.