Numerische Mathematik

Transcription

1 Numer. Mth. (2003) 95: Dgtl Object Identfer (DOI) /s Numersche Mthemtk Addtve Schwrz Methods for Ellptc Mortr Fnte Element Problems Petter E. Bjørstd 1,, Mksymln Dryj 2,, Tll Rhmn 3 1 Insttute for Informtcs, Unversty of Bergen, N-5020 Bergen, Norwy; e-ml: petter@.ub.no 2 Deprtment of Mthemtcs, Wrsw Unversty, Bnch 2, Wrsw, Polnd; e-ml: dryj@mmuw.edu.pl 3 Insttute for Mthemtcs, Unversty of Augsburg, Unverstätsstr. 14, D Augsburg, Germny; e-ml: tll.rhmn@mth.un-ugsburg.de Receved December 6, 2000 / Revsed verson receved My 17, 2002 / Publshed onlne Februry 18, 2003 c Sprnger-Verlg 2003 Summry. Two vrnts of the ddtve Schwrz method for solvng lner systems rsng from the mortr fnte element dscretzton on nonmtchng meshes of second order ellptc problems wth dscontnuous coeffcents re desgned nd nlyzed. The methods re defned on subdomns wthout overlp, nd they use specl corse spces, resultng n lgorthms tht re well suted for prllel computton. The condton number estmte for the precondtoned system n ech method s proportonl to the rto H/h, where H nd h re the mesh szes, nd t s ndependent of dscontnuous jumps of the coeffcents. For one of the methods presented the choce of the mortr (nonmortr) sde s ndependent of the coeffcents. Mthemtcs Subject Clssfcton (1991): 65N55 1 Introducton In ths pper we dscuss two vrnts of the ddtve Schwrz method for solvng systems of lgebrc equtons resultng from the pproxmton of second order ellptc problems wth dscontnuous coeffcents. The dscretzton s obtned by usng the mortr fnte element method on nonmtchng Ths work hs been supported n prt by the Norwegn Reserch Councl, grnt /420 Ths work hs been supported n prt by the Ntonl Scence Foundton, grnt NSF- CCR nd n prt by the Polsh Scence Foundton, grnt 2P03A02116

2 428 P.E. Bjørstd et l. meshes, technque whch ws frst ntroduced by Bernrd, Mdy, nd Pter n [3]. The mn chrcterstc of the method s tht the meshes on djcent subdomns cn be nonmtchng cross subdomn nterfces, llowng for the mesh on ech subdomn to be generted ndependently of the rest of the domn. The couplng between djcent subdomn meshes s done by mposng wek contnuty constrnt on the fnte element functons cross subdomn nterfces. Severl tertve solvers hve been developed for the mortr element snce t ws frst ntroduced, see for exmple [1,2, 6 8,10 12,14,17 19], nd the references theren. The frst method we ntroduce s n extenson of the orgnl verge Schwrz method of Bjørstd, Dryj, nd Vnkko [5] for second order ellptc problems usng conformng fnte elements, to the mortr cse. Ths method hs good convergence behvor, but uses computtonlly expensve corse spce. We mprove ths stuton by replcng the corse spce wth two specl corse spces, resultng n method whch s more prllel nd yet retns the essentl convergence behvor of the frst method. The bsc prncple behnd the constructon of the second method s bsed on the de used for the ddtve Schwrz method proposed by Bjørstd nd Dryj n [4]. Snce the methods re defned on nonoverlppng subdomns nd snce they contn specl corse spces, they re very well suted for prllel computtons. Ths s becuse, n ech tertve step, the lgorthms reduce to solvng severl ndependent subproblems, nd ths requres mnml communcton for the computton of the resdul. The rte of convergence of these methods when used s precondtoners for the conjugte grdent method, s proportonl to (H/h) 1/2, where H nd h re the prmeters of the corse nd the fne trngulton, respectvely. It should be ponted out here tht the convergence s ndependent of the jumps of the coeffcents. Smlrly, n the frst method the choce of mortr nd nonmortr sdes does not depend on the coeffcents of the ellptc problem. The remnder of ths pper s orgnzed s follows. In Secton 2 we recll the mortr fnte element method for our ellptc problem. We ntroduce our ddtve Schwrz methods n Secton 3 4, nd n Secton 5 we touch on some mplementton ssues. Fnlly, some numercl exmples re presented n Secton 6. 2 The dscrete problem Let = N =1 be the prtton of our computtonl domn n two dmensons, where ech s polygonl subdomn, nd the subdomns re nonoverlppng. We consder the followng dfferentl problem: Fnd u H0 1( ) such tht (1) (u,v)= f(v), v H0 1 ( ),

3 Addtve Schwrz Methods for Ellptc Mortr Fnte Element Problems 429 where nd (u, v) = N (u, v) = =1 f(v)= fvdx = N ρ ( u, v) L 2 ( ), =1 N =1 fvdx, wth ρ beng postve nd constnt. The coeffcents ρ cn however hve lrge jumps cross subdomn nterfces. We remrk tht the proposed methods cn be used s precondtoners lso for the problems where the coeffcents ρ depend on x nd re dscontnuous only cross subdomn nterfces. In ths cse the constnt ρ cn be tken s n verge of ρ (x) over the subdomn. We consder only the geometrclly conformng cse,.e., the ntersecton between the closure of two dfferent subdomns s ether empty, vertex, or whole edge. The subdomns form corse trngulton of the whole domn wth the mesh prmeter H = mx H, where H s the dmeter of. In ech subdomn, we use trngulr elements to trngulte. We ssume tht the trngles touchng the subdomn boundry re qusunform, hvng mesh sze of order h. We do not put such restrcton on the nteror trngles. We lso ssume tht the corse trngulton nd the fne trngulton n ech re shpe regulr n the sense of [9]. The resultng trngulton cn be nonmtchng cross subdomn nterfces. Let X ( ) be the fnte element spce of pecewse lner contnuous functons defned on the trngulton of nd vnshng on, nd let X h ( ) = X 1 ( 1 ) X 2 ( 2 ) X N ( N ). In order to descrbe our dscrete problem, we need the followng uxlry nottons nd fnte element spces. Let Ɣ j be n open edge common to nd j,.e., Ɣ j = j, nd let W h (Ɣ j ) nd W h j (Ɣ j ) be the restrctons of X ( ) nd X j ( j ) onto Ɣ j, respectvely. Note tht ech nterfce Ɣ j nherts two dfferent dscretztons from ts two sdes. We select one sde of Ɣ j s the mster sde, clled the mortr, nd the other sde s the slve sde, clled the nonmortr. Defne the skeleton S = ( ) \ s follows: S = m γ m, wth γ m γ n = f m n, where ech γ m denotes n open mortr edge. We wrte γ m s γ m() f t s n edge of,.e., γ m(). Let δ m = δ m(j) j be the correspondng open nonmortr edge of j tht occupes the sme geometrcl spce s γ m(),.e., γ m() = Ɣ j = δ m(j). See Fgure 1 for n llustrton.

4 430 P.E. Bjørstd et l. Ɣ j j hj h δ m(j) γ m() Fg. 1. A mortr nd nonmortr sde of subdomn nterfce wth nonmtchng grd on both sdes In ech fgure of ths pper thck lne s drwn on the mortr sde of n nterfce, whch represents the open mortr edge of tht sde. The correspondng nonmortr edge les on the other sde of the nterfce. The nonmortr edges re never shown explctly n the fgures. The end ponts of mortr or nonmortr re represented usng thck dots. Note tht ech such pont occupes the sme geometrcl spce s one of the vertces of the subdomn to whch ts edge belongs. Further, we sy tht the functon vlues (t the dscretzton ponts) on mortr re nonzero f the edge s blck nd zero f the edge s lght gry. The sme pples to the end ponts. In ddton nz (short form of nonzero) wll be used to specfy nonzeros on open nonmortr edges. We denote the set of ll vertces n by ν. As generl rule for choosng the mortrs nd the nonmortrs, we let γ m() be the mortr nd δ m(j) the correspondng nonmortr f ρ ρ j. Ths s, however, not necessry for our frst method to be presented n the next secton. Snce the trngultons on nd j my not mtch on ther common nterfce Ɣ j, see Fgure 1, the functons n X h ( ) cn be dscontnuous cross the nterfce Ɣ j. We therefore mpose wek contnuty condton cross the nterfce clled the mortr condton, see [3]. Let u h X h, where u h =u } N =1. A functon u h X h stsfes the mortr condton on δ m(j), f, for ll functons ψ M h j (δ m(j) ) (γ m() = δ m(j) = Ɣ j ), (2) (u γm() u j δm(j) )ψ ds = 0. δ m(j) Here the spce M h j (δ m(j) ) s subspce of W h j (δ m(j) ), wth functons hvng constnt vlues on elements touchng δ m(j). The condton cn be rewrtten s follows. Let m (u,trv j ) : L 2 (δ m(j) ) W h j (δ m(j) ) denote projecton defned by

5 Addtve Schwrz Methods for Ellptc Mortr Fnte Element Problems 431 (3) m (u,tr v j )ψ ds = u δm(j) ψds, δ m(j) δ m(j) for ll functons ψ M h j (δ m(j) ), nd (4) m (u,tr v j ) δm(j) = v j δm(j). Here Trv j denotes the trce of v j. Thus, u j δm(j) = m (u,trv j ) f v j = u j on δ m(j). V h s subspce of X h of functons whch stsfy the mortr condton for ll δ m S. The dscrete problem hs the form: Fnd u h =u } N =1 V h such tht (5) (u h,v h) = f(v h ), v h V h where (u h,v h ) = N (u,v ) = =1 N ρ ( u, v ) L 2 ( ), =1 nd v h =v } N =1 V h. V h s Hlbert spce wth n nner product defned by (u h,v h ). The problem hs unque soluton nd ts error bound s known, see [3]. The bss functons of the fnte element spce V h The fnte element spce V h cn be expressed s the spn of ts bss functons,.e., V h = spnφ k }, where ech bss functon φ k s ssocted wth node x k from the sets h, γ m()h (γ m() ) nd ν, for = 1,,N, except those on. h nd γ m()h denote the sets of nodl ponts of nd γ m(), respectvely. Let x () k be the locl representton of the node x k, ndctng tht the node belongs to. Let ϕ () k denote the stndrd nodl bss functon ssocted wth the node x (). The bss functons re defned s follows. k If x k h, then φ k (x) s exctly equl to ϕ () k (x). Ifx k γ m()h, then φ k (x) = ϕ () k (x) on, whle on δ m(j), where γ m() = δ m(j), φ k (x) = m (ϕ () k, 0)(x). φ k s zero t the remnng nodl ponts of j, nd zero everywhere on the remnng subdomns. For x k ν, φ k tkes ts form dependng on whether x k s pont of ntersecton between two mortrs, two nonmortrs, or mortr nd nonmortr. We defne the functon φ k for k ν below. We frst consder when x k s pont of ntersecton between mortr γ m() nd nonmortr δ n(),.e., x k γ m() δn(). Then, φ k (x) = ϕ () (x) on k

6 432 P.E. Bjørstd et l. γ m(), whle φ k (x) = m (ϕ () k, 0)(x) on δ m(j) (γ m() = δ m(j) ), nd φ k (x) = n (0,Tr ϕ () k )(x) on δ n(). Fnlly, φ k s zero t the remnng nodl ponts of nd j (γ m() = δ m(j) ), nd zero everywhere on the remnng subdomns. In the smlr wy, φ k s defned when x k s pont of ntersecton between two mortrs (γ m() nd γ n() ) or two nonmortrs (δ m() nd δ n() ). Usng the bss functons of V h, the problem (5) cn now be rewrtten n the mtrx form s (6) Au = f, where u s vector of nodl vlues of u h. The mtrx A s symmetrc nd postve defnte, nd ts condton number s of the sme order s tht of conformng fnte element method provded tht h re ll of the sme order. For the rest of ths pper we mke the followng chnges n notton. We drop the subscrpt h for functons n X h nd V h. We use the functon nme (e.g. v j ) nsted of the whole expresson contnng Trfollowed by the functon nme (.e., Tr v j ) s the second rgument to m (.,.), n other words, we smply use m (u,v j ) to denote m (u,tr v j ). 3 The ddtve verge Schwrz method In ths secton, we desgn n ddtve Schwrz method for the problem (5), whch s nspred by the ddtve verge Schwrz method of Bjørstd, Dryj, nd Vnkko n [5]. The method s defned usng the generl frmework for the ddtve Schwrz methods, see [16],.e., n terms of decomposton of the globl spce V h nto subspces nd the blner forms defned on these subspces. We decompose V h nto smller subspces V () s (7) N V h = V (0) + V (). =1 For = 1,,N, V () s the restrcton of V h to wth zero on nd on the remnng subdomns. V (0) s defned s rnge of lner opertor I A,.e., (8) V (0) =I A v : v V h }. The opertor I A s defned s follows. For u =u } N =1 V h, I A u,n,s gven by (9) (I A u )(x) = u (x), x h, u, x h,

7 Addtve Schwrz Methods for Ellptc Mortr Fnte Element Problems 433 where u = 1 (10) u γm() + u γm(j). ν γ m() δ m(),γ m(j) =δ m() Here h, s defned erler, nd h re the sets of nodl ponts of nd, respectvely, u γm() = 1 (11) u ds γ m() γ m() s the verge of u over the mortr sde γ m(), ν s the number of vertces of, nd γ m() s the length of γ m(). Note tht, wth δ m() = γ m(j) (γ m(j) j ), the verge of u over the nonmortr sde δ m(), u δm(), s equl to u γm(j) by the mortr condton. On δ m(), we lso hve (12) u (x) = m (0,u ) + m (u j, 0). We use the exct blner form for ll our subproblems,.e., for = 0,,N nd u, v V (), we defne b () (.,.) : V () V () Rs (13) b () (u, v) = (u, v). For = 1,,N, ny functon u V () s of the form u =u j } N j=1 where u j = 0 for j. Hence, for = 1,,N,wehve (14) (u, v) = (u,v ), where u =u j } N j=1 V () nd v =v j } N j=1 V (). The projecton lke opertors T () : V h V () re defned n the stndrd wy,.e., for = 0,,N nd u V h, T () u V () s the soluton of (15) b () (T () u, v) = (u, v), v V (). Note tht, for = 1,,N, the problem (15) reduces to Drchlet problem on wth homogeneous boundry condton. To compute the correspondng stffness mtrces we need the stndrd nodl bss functons ϕ () k ssocted wth x k h (nteror) only. For = 0, the problem (15) s corse spce problem, see the mplementton secton for the constructon of the stffness mtrx. Let T = T (0) + T (1) + +T (N). The problem (5) nd (6) re now replced by the followng precondtoned system, (16) Tu = g, where g = N =0 T () u. Note tht T () u cn be clculted wthout knowng u, the soluton of (5), see [16].

8 434 P.E. Bjørstd et l. 3.1 Anlyss Theorem 1 For u V h, h (17) c 0 H (u, u) (T u,u) c 1(u, u), where both c 0 nd c 1 re postve constnts ndependent of the mesh szes h = nf h nd H = mx H nd of the jumps of the coeffcents ρ. Proof. We use the generl Schwrz frmework to prove the theorem, see Chpter 5 of the book [16], where we only need to verfy three key ssumptons. Assumpton 1. V (0) s the corse spce here, nd snce V (), = 1,,N} re orthogonl wth respect to the blner form (.,.), wehveρ(e) = 1. Assumpton 2. For b (), = 0,,N,wehveω = 1 snce only the exct blner forms re used, see (13). Assumpton 1. Foru V h, let u (0) = I A u V (0) nd w = u u 0 =w } N =1. It follows mmedtely tht w = 0on. We set, for = 1,,N, u () = (0,, 0,w, 0, 0), where u () now belongs to V (). Clerly, we hve (18) u = u (0) + N u (), wth u (0) V (0) nd u () V (), = 1,,N. We need to prove tht =1 (19) N b (0) (u (0),u (0) ) + b () (u (),u () ) c H (u, u). h =1 For = 1,,N, note tht b () (u (),u () ) = (w,w ) = (u u (0),u u (0) ), where u (0) =u (0) } N =1 V (0). Hence, by the trngle nequlty, we hve N (20) b () (u (),u () ) 2(u, u) + 2(u (0),u (0) ). =1 For u (0),wehve (21) b (0) (u (0),u (0) ) = (u (0),u (0) ) = N =1 (u (0),u (0) ).

9 Addtve Schwrz Methods for Ellptc Mortr Fnte Element Problems 435 Let n be the number of nodes on. By the nverse nequlty nd the propertes of I A, we hve on ech (u (0),u (0) ) = ρ u (0) u 2 H 1 ( ) cρ h 2 K u(0) u 2 L 2 (K) (22) K B cρ (u (x) u ) 2 x h cρ h 1 u 2 L 2 ( ) + n u 2 }. The summton n the second lne s tken over ll elements K touchng the boundry, where B represents the unon of ll such elements n. Snce the subdomns re ssumed to be shpe regulr, we cn estmte the second term nsde the curly brckets s follows. (23) n u cn ν Ɣ u 2 L 2 (Ɣ) Ɣ ch 1 u 2 L 2 ( ). Here Ɣ n the summton s γ m() nd δ m(). We hve lso used the fct tht u δm() = u γm(j) on δ m() = γ m(j). Agn, for ny constnt α, we note tht u u = (u α) (u α). Usng ths, smple trce nequlty nd the Poncré nequlty (choose α to be the verge of u over the subdomn,.e., α = 1 u dx), we hve from (22) (u (0),u (0) ) cρ h 1 u 2 L 2 ( ) cρ h 1 H u 2 H 1 ( ) + H 1 u 2 L 2 ( ) c H (24) (u,u ). h Summng (24) over = 1,,N, nd usng the sum n (20) we fnlly get (19). The proof of the theorem follows. } 4 Corse reformulton In ths secton we present our second method whch s obtned from the ddtve verge Schwrz method by smply replcng ts corse spce (cll t V vg (0)) wth two new corse spces, V ( 1) nd V (0), whch cn be treted ndependently. V ( 1) s spce of functons gven by ther vlues on S, nd V (0) s specl spce hvng dmenson equl to the number of subdomns.

10 436 P.E. Bjørstd et l. The sum of these two new spces wll contn the spce V vg (0), see Remrk 2. The bsc prncple for ths pproch s bsed on the de of Bjørstd nd Dryj n [4]. The decomposton of the fnte element spce V h tkes the form (25) V h = V ( 1) + V (0) + N V (), =1 where V (), = 1,,N, re the locl subspces defned n the prevous secton. The spce V ( 1) s defned s follows. (26) V ( 1) = v V h : v(x) = 0, x h \ ( h ) }. The corse spce V (0) s defned n the followng prgrphs. Defnton 1 We sy tht node x k s connected to the subdomn f x k h. If the node x k γ m()h (x k δ m()h ) then x k s sd to be connected to both nd j f γ m() = δ m(j) (δ m() = γ m(j) ), n other words node belongng to subdomn s connected to neghbor subdomn only through the mortrs or the nonmortrs the node belongs to. As n exmple see Fgure 3(c), where the node x (k) s connected to exctly three subdomns, k, nd j only. Accordngly, for 2-D the mxmum number of subdomns node cn be connected to s three. Let χ, ssocted wth the subdomn, be the pecewse lner contnuous functon on the trngulton of, defned by ts nodl vlues t x h. For ech such node x, χ (x) = 1 j ρ j(x), where the sum s tken over the subdomns tht x s connected to, n the sense of Defnton 1. Note tht for ρ = ρ j = 1, χ s 1 for x h, 1 2 for x ( h \ ν ) nd 1 3 for x ν. We prtton the set of subdomns, = 1,,N, nto two subsets, the set of nteror subdomns N I, nd the set of boundry subdomns N B. Subdomns belongng to N B ntersect the boundry n t lest one pont, whle those of the nteror set N I do not. N B s prttoned further nto two new subsets, N BE nd N BV, where N BE s the set of subdomns tht shre t lest n edge wth, whle N BV contns subdomns tht shre only vertex wth. We ssocte wth ech subdomn the sets G nd Q contnng the ndces of ts neghborng subdomns defned s follows. G contns the ndex of neghbor j f t shres n edge Ɣ j (Ɣ j = j ) wth. Q contns the ndex of neghbor j f j s crosspont, there s

11 Addtve Schwrz Methods for Ellptc Mortr Fnte Element Problems 437 nz nz nz nz nz nz nz nz nz nz nz nz Fg. 2. Illustrton of bss functon ssocted wth n nteror subdomn, ndctng plces, lke the subdomn nteror (shded re), the mortrs (drk thck lne), the nonmortrs (mrked nz ), nd the end ponts (drk thck dots), where the functon tkes nonzero vlues subdomn k such tht Ɣ k (Ɣ k = k ) nd Ɣ jk (Ɣ jk = j k ) re the two edges of k whch ntersect t tht crosspont, nd Ɣ k s mortr n k, cf. Fgure 3(c). Two other sets, G nd Q, re lso defned n ths connecton. G s the subset of G, contnng the ndex of j for whch the edge Ɣ j touches. Q s the subset of Q, contnng the ndex of j for whch the correspondng crosspont les on. We re now redy to defne the corse spce V (0) whch s gven s the spn of ts bss functons,, = 1,,N,.e., (27) V (0) = spn : = 1,,N}. Ech functon, ssocted wth the subdomn, s functon n the fnte element spce V h. For n nteror subdomn ( N I ), see Fgure 2, the functon s constructed n three steps. We defne frst () on, then () on j for G j, nd then ()on j for Q j. () on (cf. Fgure 2) s gven s (28) 1, x h, (x) = ρ χ (x), x γ m()h ν, ρ m (χ j,χ )(x), x δ m()h,δ m() = γ m(j).

12 438 P.E. Bjørstd et l. δ m() δ m(j) δn(j) x (j) δn(j) x (j) δn(j) x (k) j j j k () (b) (c) Fg. 3. Illustrtng (ssocted wth n nteror subdomn )on j, where G j (() nd (b)) nd Q j ((c)) () on j, where G j, we hve two cses to consder. For the frst cse, let Ɣ j = δ m(j) = γ m(), see Fgure 3(). Then, on j, ρ m (χ,χ j )(x), x δ m(j)h,δ m(j) = γ m(), (29) (x) = (x), x δ n(j)h, δ n(j) δ m(j), 0, t ll other x n jh. For the second cse, let Ɣ j = γ m(j) = δ m(), see Fgure 3(b). on j s then gven s ρ χ j (x), x γ m(j)h,γ m(j) = δ m(), (30) (x) = (x), x δ n(j)h, δ n(j) γ m(j), 0, t ll other x n jh. The functon (x) n both (29) nd (30) depend on the confgurtons of nd j. Let Ɣ jk = δ n(j) = γ n(k) for some k. Both n (29) nd (30), Ɣ j Ɣ jk s nonempty. There re two dfferent stutons we need to consder for (x). For the frst, let Ɣ k be the edge such tht Ɣ k Ɣ jk s nonempty, see Fgure 4 for n llustrton. Then we hve (31) where x (j) x (j) = x (k) (x) = ρ χ j (x (j) ) n(0,ϕ (j) ) + ρ χ k (x (k) ) n(ϕ (k), 0), = Ɣ j Ɣ jk ν j nd x (k) = Ɣ k Ɣ jk ν k. Note tht (geometrclly occupyng the sme spce) snce we only consder the geometrclly conformng cse. For the other stuton, where there exsts no such Ɣ k (cf. fgures 3 3b), we smply hve (32) (x) = ρ χ j (x (j) ) n(0,ϕ (j) ). Note tht the frst term n (31) s the sme s the expresson (32), nd the second term reppers n step () below. So, wthout ny loss of generlty, we

13 Addtve Schwrz Methods for Ellptc Mortr Fnte Element Problems 439 k x (k) δ n(j) j x (j) Ɣ j Ɣ k Fg. 4. Illustrtng the stuton for (31), whch results due to hvng vertex of s cross pont of exctly three subdomns cn ssume, for the ske of smplcty, tht the stuton llustrted n Fgure 4 never occurs. The expresson for (x) s then gven by (32) only. We note lso tht the sets G nd Q re now dsjont. () on j, where Q j, s gven s follows. Let Ɣ k nd Ɣ jk be the two edges such tht Ɣ jk = δ n(j) = γ n(k) nd x (k) = Ɣ k Ɣ jk ν k, see Fgure 3(c). Then (33) (x) = ρ χ k (x (k) ) n(ϕ (k), 0), x δ n(j)h, 0, t ll other x n jh. On the remnng subdomns, = 0. Ths completes the defnton of for N I, see Fgure 2 for n exmple. If s boundry subdomn ( N B ) then the functon s defned s bove but by mposng χ j (x) = 0tx jh h for ll j N B. The vlues of on some nonmortr edges (cf. Fgure 5) touchng wll be dfferent nd therefore needs to be redefned. Below, we defne the functon on those edges. Let ˆ } N B be the set of functons constructed n exctly the sme wy s the functons ssocted wth the nteror subdomns,.e., by followng the steps ()-() bove. We use the followng rule: Any edge of N BE whch les on the boundry s mortr n. The functon, for N B, on nonmortr edges (cf. Fgure 5) touchng the boundry cn now be obtned from the correspondng ˆ. Let x () ν h nd x (j) ν j h be the end ponts (geometrclly occupyng the sme spce) of δ m() nd γ m(j), respectvely, such tht δ m() = γ m(j) j, see Fgure 5(b). At x δ m()h, we hve (cf. (28)) (34) = ˆ ρ χ (x () ) m(0,ϕ () The lst two terms bove re smply due to ϕ () ) ρ χ j (x (j) ) m(ϕ (j), 0). nd ϕ (j), respectvely.

14 440 P.E. Bjørstd et l. x () x () δ m() δ m(j) δn(j) x (j) δn(j) x (j) δn(j) x (k) j j j k () (b) (c) Fg. 5. Illustrtng ssocted wth boundry subdomn. The functon tkes zero vlues t the nodes on the outer boundry For nonmortr edges n the neghborng subdomns j, we need to consder only for G j nd Q j snce n the other cses there wll be no chnge. Consder G j. Let x() ν h nd x (j) ν j h be the end ponts (geometrclly occupyng the sme spce) of γ m() nd δ m(j), respectvely, such tht γ m() = δ m(j), see Fgure 5(). At x δ m(j)h, we hve (cf. (29)) (35) = ˆ ρ χ (x () ) m(ϕ (), 0) ρ χ j (x (j) ) m(0,ϕ (j) ). Let δ n(j) be the nonmortr referred to n (29) nd (30). (x), n ths cse, s gven by (32) only. Now let x (j) ν j, see Fgures 5()-(b).At x δ n(j)h, we then hve = ˆ ρ χ j (x (j) ) n (0,ϕ (j) ) = 0 whch follows from (32). Consder now Q j. Let δ n(j) be the nonmortr referred to n (33) wth x (k) ν k, see Fgure 5(c). From (33), t follows tht = ˆ ρ χ j (x (k)) n(ϕ (k), 0) = 0tx δ n(j)h. Consequently, s zero everywhere on j. A somewht smlr but smpler corse spce defned n terms of dscrete hrmonc functons n the context of substructurng lgorthms for the mortr fnte element problem hs been used by Dryj n [11]. Smlr spces for ddtve Schwrz methods usng conformng elements cn be found n [4, 13]. Remrk 1. The functons hve the property tht for N I, (x) + j j(x) = 1tllx, nd for N B ths equlty s true everywhere n except for the trngles touchng the boundry n t lest one pont.

15 Addtve Schwrz Methods for Ellptc Mortr Fnte Element Problems 441 Remrk 2. Let V vg (0) be the corse spce defned n the prevous secton nd let ψ =ψ } N =1 be ny functon n tht spce. Defne ψ (0) = N =1 ψ, where ψ s the verge of ψ on defned by (10), nd ψ ( 1) = ψ ψ (0). Clerly, ψ (0) V (0) nd ψ ( 1) V ( 1). The spce V ( 1) + V (0) hs hgher dmenson thn V vg (0) (0), nd hence V vg V ( 1) + V (0). As n our frst method, we use the exct blner form for ll our subproblems here. For = 1,,N, nd u, v V (), b () (.,.) : V () V () R re the blner forms defned s n (13). Let T = T ( 1) + T (0) + +T (N), be the ddtve opertor, where T () : V h V () s the orthogonl projecton n the sense of sclr product (.,.), nd T () u V () s the soluton of (15) for = 1,,N. 4.1 Anlyss Theorem 2 For u V h, h (36) c 0 H (u, u) (T u,u) c 1(u, u), where both c 0 nd c 1 re postve constnts ndependent of the mesh szes h = nf h nd H = mx H nd of the jumps of the coeffcents ρ. In order to prove the theorem we wll need some estmtes nvolvng the corse spce V (0), whch we now show n Lemm 1. Lemm 1 For u =u } N =1 V h let u (0) = =1,,N u, where u s now the verge of u over,.e., u = 1 u ds. We hve (37) (u (0),u (0) ) c H (u, u) h nd (38) N =1 ρ h 1 u (0) u 2 L 2 ( ) ch (u, u), where c s postve constnt ndependent of the mesh szes h nd H nd of the jumps of the coeffcents ρ. Proof. We prove the frst nequlty nd then show how the second nequlty cn be derved from the frst one. For the proof, we look t ech subdomn t tme, nd lso, tret the nteror nd the boundry subdomns seprtely.

16 442 P.E. Bjørstd et l. Consder frst the cse when s n nteror subdomn ( N I ). u (0) on cn be wrtten s u (0) = u + u j j. j G Q It follows from the defnton of u (0) nd Remrk 1 tht, on, u (0) u = (u j u ) j. j G Q Usng ths nd the trngle nequlty we fnd (u (0),u (0) ) = ρ u (0) u 2 H 1 ( ) (39) c ρ (u j u ) 2 j 2 H 1 ( ). j G Q We estmte the bove sum n two steps. () Estmton of the fctor (u j u ) 2 n (39). We estmte the fctor (u j u ) 2 for j G nd j Q seprtely. Let j G, nd we consder only δ m() = γ m(j) (γ m() = δ m(j) gves the sme estmte whch cn be derved usng the sme technque). Let u γm(j) nd u δm() be the verges over the mortr sde γ m(j) nd the nonmortr sde δ m(), respectvely, s defned n Secton 3. By the mortr condton, we know tht u γm(j) = u δm(). We thus hve by the trngle nequlty (40) Note lso tht (u j u ) 2 2 (u j u γm(j) ) 2 + (u u δm() ) 2}. u j u γm(j) = (u j u j ) γm(j). Usng ths nd the Cuchy-Schwrz nequlty, we estmte the frst term nsde the curly brckets of (40) s } 2 (u j u γm(j) ) 2 1 = (u j u j )ds γ m(j) γ m(j) (41) ch 1 j u j u j 2 L 2 (γ m(j) ) c u j 2 H 1 ( j ). The lst nequlty bove follows from (42) below, whch we show now. For ny constnt α, we note tht α u j = (u j α) j. Hence, by the Cuchy- Schwrz nequlty, (α u j ) 2 = 1 j j (u j α) ds } 2 ch 1 j u j α 2 L 2 ( j ).

17 Addtve Schwrz Methods for Ellptc Mortr Fnte Element Problems 443 By the trngle nequlty, the bove estmte, smple trce nequlty nd the Poncré nequlty (choose α = 1 j j u j dx), we hve } u j u j 2 L 2 ( j ) 2 u j α 2 L 2 ( j ) + α u j 2 L 2 ( j ) (42) c u j α 2 L 2 ( j ) ch j u j 2 H 1 ( j ). The second term on the rght hnd sde of (40) cn be estmted n exctly the sme wy, but ths tme we tke the verge over δ m() nsted of γ m(j). Hence, for j G, we obtn } (43) (u j u ) 2 c u 2 H 1 ( ) + u j 2 H 1 ( j ). Now consder j Q. An estmte smlr to (43) cn be derved. Let x (k) be the vertex of k such tht the two edges of k, Ɣ jk nd Ɣ k, ntersect t x (k), see Fgure 3(c). By the trngle nequlty, (44) (u j u ) 2 c (u j u Ɣjk ) 2 + (u k u Ɣjk ) 2 + (u k u Ɣk ) 2 + (u u Ɣk ) 2 }, where we hve used the sme fct whch we hve used before,.e., the verges over n nterfce from ts both sdes re the sme. Applyng the sme steps s n (41) on ech term of (44), we obtn } (45) (u j u ) 2 c u 2 H 1 ( ) + u k 2 H 1 ( k ) + u j 2 H 1 ( j ). (2) Estmton of the fctor ρ j 2 n (39). By the nverse nequlty H 1 ( ) nd the dscrete L 2 -norm, we hve (46) ρ j 2 H 1 ( ) cρ j 2 L 2 (K) cρ K B h 2 K x h 2 j (x) snce j s zero t ll x h. Hence, n order to estmte the sum n (46) we need to consder the functon j only t plces, long, where t s nonzero. Agn, we estmte the sum (fctor) for j G nd j Q seprtely. We frst consder j G. Let Ɣ j, the edge shred between nd j,be mortr γ m() n. Then, by the defnton of the bss functons (cf. (30)), the functon j on γ m() s (47) j = ρ j χ.

18 444 P.E. Bjørstd et l. Smlrly, let Ɣ j be nonmortr δ m() n. Agn, by the defnton of the bss functons (cf. (29)) nd (12), j on δ m() s (48) where x () j = ρ j m (χ j, 0) + ρ j χ (x () ) m(0,ϕ () ) + ρ jχ (x () b ) m(0,ϕ () b ), ν nd x () b ν re the two end ponts of δ m(). Now let γ n() or δ n() be nother edge n tht ntersects Ɣ j (γ m() or δ m() )tx ().Onγ n(), j s represented by the nodl vlues: j (x) = 0t ll x γ n()h except t x (), where t s equl to ρ jχ (x ()). j on δ n() wll smply be (cf. (32)) (49) j = ρ j χ (x () ) n(0,ϕ () ). Let γ o() (δ o() ) be the thrd edge n tht ntersects Ɣ j (γ m() or δ m() )t x () b. j on γ o() (δ o() ) hs the smlr representton s for γ n() (δ n() ) bove. j s zero everywhere else on. As we now know how to express j (j G ) (usng (47)-(49)) nywhere on, we estmte the sum n (46) here. To mke our clculton smple, we ssume tht γ o() (not δ o() ) ntersects Ɣ j t x () b. There re four combntons we need to nvestgte: γ m() wth γ n(), δ m() wth δ n(), γ m() wth δ n(), nd δ m() wth γ n(). Let γ m() nd γ n() ntersect t x (). Usng (47), the trngle nequlty, nd the fct tht ρ j χ (x) 1 for ll x γ m(), we obtn ρ 2 j (x) = ρ 2 j (x) cρ h 1 ρ j χ 2 L 2 (γ m() ) x h x γ m()h (50) Note tht cρ H h. ρ (ρ 2 j χ 2 (x)) = ρ j(ρ ρ j χ 2 (x)) ρ j, x γ m(), s true snce ρ ρ j χ 2(x) 1 for ll x γ m(), nd hence t s possble to replce ρ wth ρ j n the estmte (50). Let δ m() nd δ n() ntersect t x (). Note tht ρ jχ (x) 1 for ll x δ m(), ρ j χ j (x) 1 for ll x γ m(j), nd ρ j χ k (x (k) ) 1. Combnng (48)-(49), nd usng the trngle nequlty, we obtn ρ 2 j (x) = ρ 2 j (x) + ρ 2 j (x) x h x δ m()h x δ n()h cρ h 1 ρj 2 χ 2 (x() ) m(0,ϕ () ) 2 L 2 (δ m() )

19 Addtve Schwrz Methods for Ellptc Mortr Fnte Element Problems ρj 2 χ 2 (x() b ) m(0,ϕ () b ) 2 L 2 (δ m() ) + ρ j m (χ j, 0) 2 L 2 (δ m() ) } + ρj 2 χ 2 (x() ) n(0,ϕ () ) 2 L 2 (δ n() ) cρ h 1 h + h + H + h } H (51) cρ. h Here we hve used, for the frst, second nd fourth terms, the fct tht n (0,ϕ ()) 2 L 2 (δ n() ) ch, whch follows from Lemm 4 n [8]. And, for the thrd term, we hve used the fct tht m (χ j, 0) L 2 (δ m() ) c χ j L 2 (γ m(j) ), whch follows from the L 2 -stblty of mortr projectons (Lemm 1 n [10]). Agn, we note tht ρ (ρ 2 j χ 2 k (x(k) snce ρ ρ j χk 2(x(k) ) 1, nd tht nd )) = ρ j(ρ ρ j χk 2 (x(k))) ρ j ρ (ρ 2 j χ 2 (x)) = ρ j(ρ ρ j χ 2 (x)) ρ j, x δ m(), ρ (ρ 2 j χ 2 j (x)) = ρ j(ρ ρ j χ 2 j (x)) ρ j, x γ m(j) snce ρ ρ j χ 2(x) 1onδ m() nd ρ ρ j χj 2(x) 1onγ m(j) (γ m(j) = δ m() ). Hence, t s possble to replce ρ wth ρ j n the estmte (51). Fnlly, the cse when x () s the pont of ntersecton between γ m() nd δ n(),orδ m() nd γ n(), cn be treted usng combnton of the results from the frst two cses. Ths wll yeld only smlr estmtes. We now estmte the sum n (46) for j Q. Let x (k) be vertex of k such tht the two edges of k, Ɣ jk nd Ɣ k, ntersect t x (k), nd tht Ɣ k s mortr (γ n(k) )n k. Let δ n() be the correspondng nonmortr n. We use (33) to represent j on δ n() s (52) j = ρ j χ k (x (k) ) n(ϕ (k), 0). Note tht ρ j χ k (x (k) ) 1. By Lemm 5 n [8], we thus hve ρ 2 j (x) = ρ 2 j (x) x h x δ n()h (53) cρ h 1 ρj 2 χ k 2 (x(k) ) n(ϕ (k), 0) 2 L 2 (δ n() ) cρ h 1 h cρ.

20 446 P.E. Bjørstd et l. In the sme wy s before, snce ρ ρ j χk 2(x(k) ) 1, we cn esly wrte ρ ρj 2χ k 2(x(k) ) ρ j, whch llows us to replce ρ n (53) wth ρ j. Note tht we lso need to be ble to replce ρ n (53) wth ρ k, see (45). Ths s possble snce γ n(k) = Ɣ k s mortr n k, whch mples tht ρ ρ k. Snce the subdomns re shpe regulr, the mxml number of neghbors n G Q cn be bounded ndependently of the totl number of subdomns, N. Now, combnng (39) (53), for N I, we obtn (54) (u,u ) + (u (0),u (0) ) c H h j (u j,u j ). j G Q The bove estmte pples lso to the cse when s boundry subdomn ( N B ), whch we show now. We strt by ntroducng n uxlry functon û (0) = N I u + N B u ˆ. The functon ˆ for N B hs been defned before, see on pge 439. On, where N B, the functon û (0) tkes the form û (0) = u + u j j + u j ˆ j. j G Q, j N I j G Q, j N B By the trngle nequlty, we fnd (55) (u (0),u (0) ) 2 (u (0) û (0),u (0) û (0) ) + (û (0), û (0) ) }. The functon û (0) on hs the sme form s u (0) on n nteror subdomn. The second term hs, therefore, the sme bound s n (54). For the frst term we note tht (u (0) û (0) ) = u ( ˆ ) + u j ( j ˆ j ) j G Q on. Usng ths relton nd the trngle nequlty we hve (u (0) û (0),u (0) û (0) ) (56) = ρ u (0) û (0) 2 H 1 ( ) c ρ u 2 ˆ 2 H 1 ( ) + } ρ u 2 j j ˆ j 2 H 1 ( ). j G Q We estmte the rght hnd sde n two steps gn. () Estmton of u 2 nd u2 j n (56). Both nd j here re boundry subdomns. We derve estmtes for only u 2 snce, for u2 j, the estmtes re then strghtforwrd.

21 Addtve Schwrz Methods for Ellptc Mortr Fnte Element Problems 447 cn ether be n N BE or n N BV.For N BE, we use smple trce nequlty nd Fredrchs nequlty (u s zero on boundry edge of )to get (57) u 2 = } 1 2 u ds ch 1 u 2 L 2 ( ) c u 2 H 1 ( ). For N BV, we get n extr H h fctor n the estmte. Note tht u s zero t lest t one pont on, nd hence, for ny constnt α, α mx x u (x) α. Usng ths, smple trce nequlty nd the Poncré nequlty (choose α = 1 u dx), we obtn u 2 4 mx x u (x) α 2 ch 1 u α 2 L 2 ( ) (58) c H h u 2 H 1 ( ). (2) Estmton of the two semnorms n (56). We estmte the frst nd the second semnorms seprtely. By the nverse nequlty nd the dscrete L 2 - norm, for the frst semnorm, we hve ρ ˆ 2 H 1 ( ) cρ h 2 K ˆ 2 L 2 (K) cρ ( (x) ˆ (x)) 2 K B x h = c ρ ˆ 2 (x) + (59) ρ ( (x) ˆ (x)) 2 x h h δ m() x δ m()h }. The outer sum n the second term bove s tken over those nonmortr sdes of, tht touch the boundry t most t two ponts. We hve used the fct tht (x) = ˆ (x) t ll x h, x γ m()h \ h nd x Ɣ j where j s n nteror subdomn (.e., j N I ). For smplcty, we ssume tht δ m() n (59) touches t only one pont, snce the estmtes for the cse where δ m() touches the boundry t two ponts wll smply contrbute fctor 2 to the constnts n the estmtes. For N BE nd N BV, the frst term n (59) cn be bounded by H cρ h nd cρ, respectvely. In ether cse, for the second term, we use (34), the trngle nequlty, nd Lemm 4 nd Lemm 5 n [8], to get

22 448 P.E. Bjørstd et l. ρ ( (x) ˆ (x)) 2 x δ m()h cρ h 1 ρ 2 χ 2 (x() ) m(0,ϕ () ) 2 L 2 (δ m() ) + ρ 2 χ j 2 (x(j) ) m(ϕ (j), 0) 2 L 2 (δ m() ) cρ h 1 h + h } (60) cρ. Here x () ν h nd x (j) ν j h (geometrclly occupyng the sme spce) re the end ponts of δ m() nd γ m(j), respectvely. We hve used the fct tht ρ χ (x ()) 1 nd ρ χ j (x (j) ) 1. For the second semnorm n (56), we derve estmtes for only N BV snce, for N BE, t s strghtforwrd to see tht the sme estmtes hold. By the nverse nequlty nd the dscrete L 2 -norm, we hve ρ j ˆ j 2 H 1 ( ) cρ j ˆ j 2 L 2 (K) (61) K B cρ ( j (x) ˆ j (x)) 2 x h snce j (x) ˆ j (x) = 0 t ll x h. Agn, we estmte the sum for j G nd j Q seprtely. Consder frst j G. Let x() ν h nd x (j) ν j h (geometrclly occupyng the sme spce) be the end ponts of δ m() nd γ m(j), respectvely, such tht δ m() = γ m(j), or the end ponts of γ m() nd δ m(j), respectvely, such tht γ m() = δ m(j). We ssume gn tht they touch only t one pont. Let δ n() or γ n() be the other edge of tht ntersects δ m() or γ m() t x (). As before, there re four cses we need to nvestgte dependng on whether the two edges of re mortr or nonmortr. As the frst cse, let x () ν h be the pont of ntersecton between the two mortrs γ m() nd γ n(). ˆ j (x) j (x) = 0 t ll x h except t x (), where t s equl to ˆ j (x ()) = ρ jχ (x () ). Hence (62) ρ ( j (x) ˆ j (x)) 2 = ρ ρj 2 χ 2 (x() ) ρ j x h usng the fct tht ρ ρ j χ 2(x() ) 1. As the second cse, let x () ν h be the pont of ntersecton between the two nonmortrs δ m() nd δ n(). ˆ j (x) j (x) = 0 t ll x h except t x (δ m()h δ n()h ) nd x ().Atx δ m()h, followng (35), ˆ j j cn be wrtten s ˆ j j = ρ j χ j (x (j) ) m(ϕ (j), 0) + ρ jχ (x () ) m(0,ϕ () ), }

23 Addtve Schwrz Methods for Ellptc Mortr Fnte Element Problems 449 nd t x δ n()h, ˆ j j, by defnton (cf. (32)), hs the form ˆ j j = ˆ j = ρ j χ (x () ) n(0,ϕ () ). It follows from the trngle nequlty, Lemm 4 nd Lemm 5 n [8], tht ρ ( j (x) ˆ j (x)) 2 x h = ρ ρj 2 χ 2 (x() ) + ρ (63) ρ j + cρ h 1 x δ m()h δ n()h ( j (x) ˆ j (x)) 2 ρj 2 χ j 2 (x(j) ) m(ϕ (j), 0) 2 L 2 (δ m() ) + ρj 2 χ 2 (x() ) m(0,ϕ () ) 2 L 2 (δ m() ) + ρj 2 χ 2 (x() ) n(0,ϕ () ) 2 L 2 (δ n() ) ρ j + cρ j h 1 h + h + h } cρ j. We hve used tht ρ ρ j χj 2(x(j) ) 1 nd ρ ρ j χ 2(x() ) 1. For the other cse, where x () ν h s pont of ntersecton between ether γ m() nd δ n(),orδ m() nd γ n(), smlr estmtes cn be derved by smply combnng the results from the frst two cses. We now estmte the sum n (61) for j Q. Let x(k) ν k be the pont of ntersecton between the two edges Ɣ jk nd Ɣ k of k, for some k, such tht Ɣ k = δ n() = γ n(k). ˆ j (x) j (x) = 0 t ll x h except t x δ n()h.atx δ n()h, ˆ j (x) j (x), by defnton (cf. (33)), hs the form ˆ j j = ˆ j = ρ j χ k (x (k) ) n(ϕ (k), 0). By the nverse nequlty, the dscrete L 2 -norm, Lemm 5 n [8] nd the fct tht ρ ρ j χk 2(x(k) ) 1, we then obtn ρ ( j (x) ˆ j (x)) 2 = ρ ( j (x) ˆ j (x)) 2 x h x δ n()h (64) cρ h 1 ρj 2 χ k 2 (x(k) ) n(ϕ (k), 0) 2 L 2 (δ n() ) cρ j h 1 h = cρ j. Now, combnng the nequltes (56)-(64), nd replcng the resultng estmte together wth the estmte (the sme s n (54)) for (û 0), û 0) ) n (55), we obtn the sme estmte s (54). The proof of (37) now follows by summng (54) over ll subdomns. }

24 450 P.E. Bjørstd et l. Dervton of the nequlty (38). The second nequlty of the lemm follows from the proof of the frst prt. Let N I. We then hve ρ u (0) u 2 L 2 ( ) c ρ (u j u ) 2 j 2 L 2 ( ), j G Q where we hve replced the H 1 -semnorm wth the L 2 -norm n (39). The estmte for j 2 L 2 ( ) wll gn fctor h2 compred wth j 2 H 1 ( ), see (50), (51) nd (53). Ths results n fctor h H nsted of H h.ifwenow follow the sme steps s n the proof of the frst prt of the lemm we obtn ρ u (0) u 2 L 2 ( ) ch H (u,u ) + (65) j (u j,u j ). j G Q For N B we use exctly the sme rguments, whch gn leds to the fctor h H nsted of H h, nd follow the sme steps s n the proof of the frst prt of the lemm for N B to get the sme estmte s (65). The proof of (38) thus follows by summng (65) over ll subdomns. The proof of Lemm 1 s now complete. We re now redy to prove our theorem. Proof (Theorem 2). We use the generl Schwrz frmework to prove the theorem, nd, s before, we hve three key ssumptons to verfy. Assumpton 1. ρ(e) = 1 snce the subspces V (0) nd V ( 1) re the two corse spces, see [16]. Assumpton 2. Exct blner forms re used for ll subproblems, hence, ω = 1 for = 1,,N. Assumpton 1. Foru =u } N =1 V h, we defne u (0) s u (0) = u, =1,,N where u = 1 u ds nd, = 1,,N re the bss functons of V (0). Let w = u u (0) =w } N =1. Let u( 1) =u ( 1) } N =1 V ( 1) be defned on ech by the nodl vlues s follows u ( 1) (x) = w (x), x h, 0, x h. Smlrly, let u (j) =u (j) } N =1 V (j),j = 1,,N be defned by the nodl vlues s u () w (x), x = h, 0, x ( h \ h ), wth u (j) = 0 whenever j. Clerly, u () V () nd u = N = 1 u(). Usng the bove splttng of u, we need to show tht

25 Addtve Schwrz Methods for Ellptc Mortr Fnte Element Problems 451 (66) N b () (u (),u () ) c H (u, u). h = 1 For u ( 1) =u ( 1) } N =1 V ( 1),wehve (67) b ( 1) (u ( 1),u ( 1) ) = N =1 (u ( 1),u ( 1) ). By the nverse nequlty, the dscrete L 2 -norm, the trngle nequlty nd the defnton of u ( 1),weget (u ( 1),u ( 1) ) = ρ u ( 1) 2 H 1 ( ) cρ (68) K B h 2 K u( 1) 2 L 2 (K) cρ w 2 (x) x h cρ h 1 u u (0) 2 L 2 ( ) cρ h 1 u u 2 L 2 ( ) + u(0) u 2 L 2 ( ) For the frst term nsde the curly brckets we hve (69) u u 2 L 2 ( ) ch u 2 H 1 ( ), whch follows from (42). For the second term, we note, by the defnton of u (0), tht u (0) (x) = u t every x h, nd hence u (0) u 2 L 2 ( ) ch (u (0) (x) u ) 2 x h ch h 2 K u(0) u 2 L 2 (K) K B (70) ch 1 u (0) u 2 L 2 ( ). Now replcng (69) nd (70) n (68), ddng over = 1,,N, nd fnlly usng Lemm 1 on the second term, we obtn }. (71) b ( 1) (u ( 1),u ( 1) ) c H (u, u). h From Lemm 1 t follows mmedtely tht (72) b (0) (u (0),u (0) ) c H (u, u). h Fnlly,

26 452 P.E. Bjørstd et l. N b () (u (),u () ) = =1 c N =1 N =1 (u (),u () ) (u, u) + (u (0),u (0) ) + (u ( 1),u ( 1) ) = c (u, u) + b (0) (u (0),u (0) ) + b ( 1) (u ( 1),u ( 1) ) } } (73) c H (u, u). h The nequlty (66) now follows from (71), (72) nd (73). Ths concludes the proof of Theorem 2. 5 Implementton ssues The subproblems ssocted wth the subspces V (), = 1,,N, re the stndrd locl subproblems, t s well known how to mplement ther solvers. In ths secton we brefly dscuss the corse problems only. The bss functons φ k } of the fnte element spce V h re ssocted wth the subdomn nteror nodes h, the mortr nodes γ m()h (γ m() ) nd the subdomn vertces ν, = 1,,N, except those on, see Secton 2 for descrpton. Let u be the vector representton of u V h wth respect to the bss of V h, whch contns the nodl vlues of u. The mtrx A of the dscrete problem cn be splt s follows. Let λ be the unon of h, γ m()h where γ m(), ν nd γ m(j)h where γ m(j) = δ m(), except those nodes on. Let R be the restrcton mtrx contnng only ones nd zeros, whch, f multpled wth u from the left, wll return vector of length λ contnng the nodl vlues of u only t the nodes of λ. Let R T be the correspondng extenson opertor. Now let A be the locl stffness mtrx ssocted wth the subdomn, whose rows nd columns correspond to the nodes of the set λ. Usng the bove three types of mtrces we cn splt the globl stffness mtrx s A = N =1 RT A R, where A = (φ j,φ l )} wth x j,x l λ. Note tht A, = 1,,N cn be constructed loclly by the processor (vrtul processor) responsble for. We use these locl mtrces to construct our corse stffness mtrces. The method wth the corse reformulton Consder frst the method wth the corse reformulton. Let V cr ( 1) nd V cr (0) be the two corse spces s defned n the prevous secton. We use the mortr bss of V h to represent the bss functon of V cr (0) (cf. (27)) s = k (x k )φ k, where (x k ) s the nodl vlue of t node x k.by the defnton of, the nodl vlue t x k wll be

27 Addtve Schwrz Methods for Ellptc Mortr Fnte Element Problems 453 (74) 1, x k h, ρ χ (x k ), x k γ m()h ν, (x k ) = ρ χ j (x k ), x k γ m(j)h,γ m(j) = δ m(), ρ χ j (x k ), x k δ m(j)h,δ m(j) = γ m(), 0, x k h, 0, otherwse. Let v be the vector representton of wth respect to the bss of V h. Let A (0) cr be the correspondng stffness mtrx of sze N N, then the element (A (0) cr ) jl s gven by ( j, l ) = v T j Av l = N =1 (R v j ) T A (R v l ). Note tht the term nsde the summton s nonzero when both j nd l belong to G Q } s both j nd l n tht cse wll hve nonzero support on. The mtrx A (0) cr cn thus be ssembled n the stndrd wy from ts element mtrces A (0), = 1,,N, where A (0) contns elements from the set (R v j ) T A (R v l )} j,l, where j,l G Q }. Note here tht the vectors v j nd v l both hve constnt vlues s ther elements correspondng to the nteror nodes h,.e., one f the ndces re equl to nd zero otherwse. Ths nformton cn be used to clculte the elements cheply, for nstnce, by tretng the whole set of nodes h s sngle node n the clculton. Now, for the spce V cr ( 1), we note tht t s smply spnφ l : x l S h }, where S h s the set of ll mortr nodes nd the vertces of subdomns, = 1,,N, except those on, see (26). The constructon of the correspondng stffness mtrx A ( 1) cr s qute strghtforwrd, nd cn be extrcted from the smll mtrces A wthout ny extr cost. The verge method Consder now the verge method. Let V vg (0) (0) be ts corse spce. V vg cn be expressed usng set of some bss functons, whch we denote s l }. Note tht the dmenson of ths corse spce s the sme s tht of V cr ( 1) bove. We defne the bss functons s follows: Assocte ech functon l to node x l from the set S h by settng the functon vlue to 1 t x l nd 0 t ll other nodes n S h, then, by the defnton of V vg (0), see (8)-(11), the nodl vlue t ny node x k s gven s 1, x k = x l, 0, x k (S h \x l }), 1 ν m, x k h,x l γ m()h, 1 ν l (x k ) = j m, x k jh,x l γ m()h,γ m() = δ m(j), (75) 0, x k h,x l = δ m() δ n(), 1 ν m, x k h,x l = γ m() δ n(), 1 ν ( m + n ), x k h,x l = γ m() γ n(), 0, otherwse.

28 454 P.E. Bjørstd et l. Here m s the verge over the edge γ m(), see (11), of functon n W h (γ m() ) hvng one s ts vlue t the node x l γ m()h nd zeros t the other nodes. Smlrly, n defnes the verge over the edge γ n(). l, n terms of the mortr bss, cn be wrtten s l = k l(x k )φ k. For nstnce, for x l γ m()h, where γ m(), l (x) = φ l (x) + 1 ν m ϕ k (x) + 1 ν x k j m h x k jh ϕ k (x). We thus hve V vg (0) = spn l : x l S h }. The constructon of the correspondng stffness mtrx A (0) vg s now smlr to tht of A(0) cr. Snce the corse spces do not need ny corse grd dscretzton, the methods re esly pplcble to unstructured grds. The method wth the corse reformulton, however, hs some computtonl dvntges over the verge method. We remrk tht A (0) vg hs much lrger stencl thn A ( 1) cr, even though they hve the sme dmenson, whch mkes A (0) vg much denser mtrx thn A ( 1) cr. By creful orderng of the nodes, t s possble to solve the A ( 1) cr corse problem by frst elmntng the unknowns ssocted wth ll mortr nodes, whch s done by solvng sequence of trdgonl systems, nd then by solvng Schur complement system on the subdomn vertces. In ddton, A (0) cr hs mnml dmenson. Solvng the two corse problems of the method wth the corse reformulton s therefore computtonlly cheper thn solvng the corse problem of the verge method. Note, n ddton, tht the two corse problems cn be solved n prllel. 6 Numercl results In ths secton, we present results from the numercl experments crred out n Mtlb usng the two vrnts of the ddtve Schwrz methods, the verge method nd the method wth the corse reformulton, ntroduced n ths pper. We hve seen n the prevous sectons tht the methods use only the exct blner forms for ther subproblems. Two vrnts of the verge method whch use dfferent pproxmte blner forms for ther corse spces cn be found n [15]. For the present pper, we tke only the best pproxmte vrnt nd compre t wth the exct vrnts. The CG method s used to solve the precondtoned system (16). The terton stops when the resdul norm s reduced by the fctor We choose our model problem to be defned on the unt squre nd wthout loss of generlty, we let the soluton hve zero boundry vlues. The force functon f s chosen to hve the form f(x)= 2π 2 sn(πx 1 ) sn(πx 2 ). We use the followng choce of mortr nd nonmortr sdes: If ρ ρ j then choose the mortr sde to be n,.e., γ m() = δ m(j) (γ m() nd δ m(j) j ).

29 Addtve Schwrz Methods for Ellptc Mortr Fnte Element Problems 455 Tble 1. Comprson of the three vrnts of the ddtve Schwrz methods. Iterton counts nd condton number estmtes (n prentheses) re shown for fxed rto H/h Subdomns Averge wth Averge wth Corse pprox. blner form exct blner form reformulton (20.80) 27 (11.41) 29 (13.46) (21.13) 29 (11.62) 32 (13.96) (21.08) 30 (11.55) 34 (14.41) Tble 1 shows terton counts nd condton number estmtes for dfferent prttons of the domn usng the two verge methods, the pproxmte nd the exct vrnt, s well s the method wth the corse reformulton. In ll tests the subdomns re trngulted usng, n checkerbord order, ether 50 or 72 rght ngle trngles of the sme sze, so tht ny two neghborng subdomns get nonmtchng grd cross ther nterfce nd the rto H/h remns fxed. For fxed prtton, the coeffcents ρ re pcked unformly from the ntervl [10 1, 10 3 ] nd then dstrbuted rndomly mong the subdomns. As we cn see from the tble, the exct vrnt of the verge method hs the best condton number estmtes, whle the correspondng estmtes for the pproxmte vrnt re bout twce s bg. It should be noted tht the sze of ther corse spce problems re the sme, nd the correspondng stffness mtrces hve the sme densty pttern. So unless n effcent lgorthm s found for the pproxmte verson ths wll not be n ttrctve method. The exct verson, however, wth ts very hgh computtonl complexty, cnnot be consdered prctcl. The method wth the corse reformulton, on the other hnd, hs condton number estmtes whch re close to those of the exct vrnt of the verge method. If we tke nto ccount the computtonl complexty nd the prllel propertes of the lgorthms we cn conclude tht the method wth the corse reformulton s n ttrctve method. The condton number estmtes, s seen from Tble 1, remn close to constnt for fxed H/hsuggestng tht the condton number depends only on the rto H/h. In Tble 2, we see tht the condton number estmtes become pproxmtely doubled s we double the rto H/hby movng from left to rght n the tble, mplyng tht the dependence s lner. The subdomn sze H nd the mesh sze h n the frst column of ths tble correspond to the sme prtton nd dscretzton whch re used for the second row of Tble 1,.e. 8 8 = 64 subdomns ech hvng ether 50 or 72 trngles. The coeffcents ρ re chosen n the sme wy s before, but due to dfferent (rndom) dstrbuton of the coeffcents the condton number estmtes here re dfferent from the correspondng estmtes n Tble 1.

30 456 P.E. Bjørstd et l. Tble 2. Iterton counts nd condton number estmtes (n prentheses) llustrtng the lner dependence of the condton number on the rto H/h Method Subdomn sze, Mesh sze H,h} H,h/2} 2H,h/2} Avg., exct 29 (11.58) 46 (28.06) 66 (62.12) Corse ref. 32 (14.40) 48 (30.84) 67 (65.80) Tble 3. Iterton counts nd condton number estmtes (n prentheses) llustrtng the nsenstveness of the condton number to the coeffcent jump Method Coeffcents ρ, checkerbord dstrbuton 1, 10 0 } 1, 10 2 } 1, 10 4 } Avg., exct 23 (10.37) 26 (11.70) 26 (11.73) Corse ref. 27 (13.88) 29 (14.19) 30 (14.24) An mportnt feture of the methods s tht they re ll nsenstve to jumps n the coeffcent ρ cross the subdomn boundres. Ths cn be seen from Tble 3, where checkerbord dstrbuton of the coeffcents s used. As we move from one column to nother, n ths tble, the jump s ncresed or decresed by the fctor 10 2, but the condton number estmtes remn unffected. 64 subdomns, ech contnng ether 50 or 72 trngles, re used for the tests. The numercl results of ths secton smply confrm our theoretcl results. Further experments wth these methods hve confrmed tht the verge methods, both the exct- nd the pproxmte- vrnt, show convergence whch s ndependent of the choce of the mortr nd the nonmortr sdes, whle the method wth the corse reformulton requres tht γ m() = δ m(j) f ρ ρ j. References 1. Achdou, Y., Kuznetsov, Y.A.: Substructurng precondtoners for fnte element methods on nonmtchng grds. Est-West J. Numer. Mth. 3, 1 28 (1995) 2. Achdou, Y., Mdy, Y., Wdlund, O.B.: Itertve substructurng precondtoners for mortr element methods n two dmensons. SIAM J. Numer. Anl. 36, (1999) 3. Bernrd, C., Mdy, Y., Pter, A.T.: A new non conformng pproch to domn decomposton: The mortr element method, n Collège de Frnce Semnr, H. Brezs nd J.-L. Lons, eds., Ptmn, Appered s techncl report lredy n Bjørstd, P.E., Dryj, M.: A corse spce formulton wth good prllel propertes for n ddtve schwrz domn decomposton lgorthm, Submtted to Numersche Mthemtk, (1999)