RESTRICTED ADDITIVE SCHWARZ METHOD WITH HARMONIC OVERLAP

Transcription

1 RESTRICTED ADDITIVE SCHWARZ METHOD WITH HARMONIC OVERLAP XIAO-CHUAN CAI, MAKSYMILIAN DRYJA, AND MARCUS SARKIS Abstract. In ths paper, we ntroduce a new Schwarz precondtoner and wth a new coarse space. We construct the precondtoner by groupng together, n one precondtoner, features from the addtve overlappng Schwarz methods, from teratve substructurng methods and from a class of restrcted addtve Schwarz methods. The precondtoner s symmetrc and consdered as a symmetrzed verson of restrcted addtve Schwarz precondtoners. We also enhance the precondtoner wth a new coarse space whch s smple, easly parallelzable, has smaller stencl, and has one degree of freedom per substructure and can be used n problems wth unstructured meshes. We study the spectral bounds for the method and dscuss the several advantages of ths precondtoners. Numercal results theory wll be provded. Key words. Schwarz precondtoner, fnte element, overlappng doman decomposton, ellptc equatons, coarse space AMS(MOS) subject classfcatons. 65N30, 65F0. Introducton. Through out ths paper, C and c, are postve generc constants that are ndependent of any of the mesh parameters and the number of subdomans. All the domans and subdomans are assumed to be open,.e., boundares are not ncluded n ts defnton. 2. Dscretzaton. Fnd u H 0 (Ω), such that (2.) where a(u, v) = Ω a(u, v) = f(v) v H0 (Ω), u v dx and f(v) = fv dx for f L 2 (Ω). Ω For smplcty, let Ω be a bounded polygonal regon n R 2 wth a dameter of sze O(). The extenson of the results to R 3 can be carred out easly by usng the theory developed here n ths paper and the well-known three-dmensonal addtve Schwarz technques; [5, 6, 7, 4]. We ntroduce a trangulaton T h (Ω) whch s shape regular and for smplcty quas-unform of sze h. We then defne a fnte element space V(Ω), whch we denote also by V, whch s the space of contnuous pecewse lnear functons assocated wth the trangulaton T h (Ω) and whch vanshes on Ω, the boundary of Ω. We are nterested n solvng the followng dscrete problem assocated to (2.): Fnd u V such that (2.2) a(u, v) = f(v), v V. 3. Notatons and Defnton of Subspaces. Let W be the set of the n nteror nodes of T h (Ω). We assume that the graph parttonng has been appled and has resulted n N nonoverlappng subsets W 0, =,..., N whose unon s W. We defne the overlappng partton of W as follows. Let {W } be the one-overlap partton of Dept. of Comp. Sc., Unv. of Colorado, Boulder, CO 80309, ca@cs.colorado.edu. Faculty of Math. Info. and Mech., Warsaw Unv., Warsaw, dryja@mmuw.edu.pl. Mathematcal Scences Department, Worcester Polytechnc Insttute, Worcester, MA msarks@wp.edu

2 W, where W W 0 s obtaned by ncludng all the mmedate neghborng vertces of the vertces n W 0. Usng the dea recursvely, we can defne a δ-overlap partton of W, W = N W δ, = where W δ W 0 wth δ levels of overlaps wth ts neghborng subdomans. Here δ 0 s an nteger, and δh s the lenght of the extenson. We defne regons Ω R by the unon of all the elements of T h (Ω) whch have all the three vertces on W 0 Ω. We denote H the representatve sze of the subregon Ω R. We note that we can always defne a subregon nduced by a set of nodes. Assume for nstance that a set Z s a subset of W WB 0. Here W B 0 s the set of boundary nodes. We denote nduced subregon of Z by Ω(Z) defned as the unon of Z, unon the open elements (trangles) of T h (Ω) whch have at least one vertex of Z, and unon of the open edges of these trangles whch has at least one endpont as a vertex of Z. we note that Ω(Z) s always an open regon f Z s a subset of W. We defne the extended regon Ω δ by Ω(W δ). We ntroduce the space V V H0 (Ω δ ) extended by zero outsde Ω\Ωδ. It s easy to check that we can decompose V as V = V + V V N. Ths decomposton wll be used n defnng the regular addtve Schwarz algorthm wthout a coarse space [3, 0]. In order to defne the new method,.e. the restrcted addtve Schwarz method wth harmonc overlap wth the new coarse space, we have to ntroduce an mportant dea of how we deal wth boundary Drchlet data. We note that we do not need to examne dfferent geometrc cases as they appear n the studes of teratve substructurng methods. There, t s needed to consder geometrcal cases such as f a subdoman touches the boundary n one pont or a whole edge; or f a subdoman should have a degrees of freedoms assocate to the coarse space. Ths consderaton also appears n the regular addtve Schwarz method wth an exotc coarse spaces [2, 5,,?, 7]. Let Ω 0 B = Ω(W B 0 ),.e. one layer of elements near the boundary Ω. We note that Ω belongs to Ω B. We also defne W ˆδ B... WB W B 0 wth ˆδ levels of extenson by addng recursvely neghborng vertces. And defne Ωˆδ B = Ω(W ˆδ B ). We note that ˆδ s gong to be same order as δ. The reason to choose ˆδ possbly dfferent to δ s because the overlap between Ωˆδ B and a Ω δ next to the boundary s (ˆδ, whle the overlap between two floatng subdomans are (2δ. Therefore, choosng ˆδ = 2δ may be more approprate than choosng ˆδ = δ. We wll see ths clearly n the analyss. We treatment of mxed Drchlet and Neumann boundary condtons s also trval. The only thng we need to do s to move de boundares assocated to Neumann part from WB 0 to W. We wll not treat ths case n the paper snce no new deas are necessary. We note that the coarse space that we ntroduce n ths paper, one degree of freedom per subdoman s used ndependent f t s a Drchlet or Neumann data, or how the subdoman touches the boundary. Let Γ = Ω δ \ Ω,.e., the part of the boundary of Ωδ that does belong to the Drchlet boundary part. Also let Γ B = Ωˆδ B \ Ω. We defne the nterface overlappng boundary Γ as the unon of all the Γ and Γ B,.e. Γ = Γ B ( N = Γ ). We then defne the followng subset of nodes of W : 2

3 W Γ W Γ (the nterface nodes) W Γ W Γ W δ (the local nterface nodes) W,n Γ W Γ W 0 (the local nternal nterface nodes) W,cut Γ W Γ\W,n Γ (the local cut nterface nodes) W,ovl δ (W δ\w Γ) (( j W j δ) W ˆδ B ) (the local overlappng nodes) W,non δ W δ\(w Γ W,ovl δ ) (the local nonoverlappng nodes) W,n δ W,non W,n Γ (the nternal nodes) We remark that the sets W,n Γ form a nonoverlappng decomposton of W Γ and t wll play a very mportant role n the defnton of the new Schwarz methods and n the new coarse space ntroduced later n ths paper. Let x k W be a mesh pont and φ xk V the fnte element bass functon assocated wth x k,.e. φ xk (x k ) =, and φ xk (x j ) = 0, j k. We say u V s dscrete harmonc at x k f a(u, φ xk ) = 0. If u s dscrete harmonc at a set of nodal ponts Z, we say u s dscrete harmonc n Ω(Z). We defne Ṽ as a subspace of V consstng of functons that vansh on the cutng nodes W,cut Γ and dscrete harmonc at the nodes W,ovl δ. We defne Ṽ as a subspace of V defned as Ṽ = Ṽ + Ṽ2 + + ṼN. It s easy to see that the sum above s a drect sum. We defne P : Ṽ Ṽ to be the projecton operators such that, for any u Ṽ (3.) a( P u, v) = a(u, v), v Ṽ. The restrcted addtve Schwarz wth harmonc overlap method (RASH) s defne as P = P. We also defne P : V V by = (3.2) a(p u, v) = a(u, v), v V. The regular addtve overlappng Schwarz method (AS) s defned as P = P. = Equvalentelly, we can defne P and P n matrx notatons. Let us defne W δ W δ\w,cut Γ, and let Ω( W δ) be the nduced doman. It s easy to see that Ω( W δ ) s the same as Ω δ but wth cuts. We denote Ω( W δ) by Ω δ. We have then Ṽ = V H0 ( Ω δ ) and dscrete harmonc on Ω(W,ovl δ ). Let R δ : W W δ be restrcton operator assocated to the the regular pontwse nterpolaton from V to Ṽ. And R δ : W W δ the restrcton operator from V to V. 3

4 Then P u, for u Ṽ n matrx notaton s defned as follows: and P u, for u V s defned as where P u = P u = ( R δ ) T (Ãδ ) Rδ Au = (R δ ) T (A δ ) R δ Au = Ã δ = R δ A( R δ ) T A δ = R δ A(R δ ) T. 4. A New Coarse Space. We next ntroduce a new coarse space Ṽ0. The assocated coarse problem wll be then added to P. Ṽ 0 s a subspace of Ṽ and t s defned by the followng coarse bass functons { f node φ xk W,n δ 0(x k ) = 0 f node x k W \ W δ and φ 0 s dscrete harmonc at x k W,ovl δ. We note that support of φ 0 s n Ω δ and therefore ts constructon can be performed n parallel and wth no communcaton. We remark also that the coarse space Ṽ 0 can also be used as a coarse space for the regular addtve Schwarz method snce Ṽ 0 s a subspace of Ṽ, and Ṽ a subspace of V 0. An nterpolaton-lke operator I 0 = N = I 0, where the I0 : V Ṽ s defned as follows: (I 0u)(x) = ū φ 0(x). Here ū = Ω δ udx. Ω δ The value ū s the average of u on the extended regon Ω δ. Here Ωδ s the area of the regon Ω δ. We have then (I 0 u)(x) = = ( Ω δ Ω δ udx)φ 0(x). We defne the coarse space Ṽ0 as the range of I 0, or equvalently, as a lnear combnatons of the φ 0. We ntroduce P 0 : V Ṽ0 as the operator such that, for any u V, (4.) a( P 0 u, v) = a(u, v), v Ṽ0 4

5 Let R 0 T : Ṽ0 V be the standard embedded prolongaton operator. The coarse projecton operator P 0 n matrx notaton s defned by P 0 = R 0 T Ã 0 R 0 A where Ã0 = R 0 A R 0 T. We remark that Ã0(, j) = 0 f domans and j are not neghbors. We note that Ã0 s more sparse than coarse spaces matrces that appear n other methods such as Neumann-Neumann [9] and FETI [8] precondtoners. Another mportant feature of ths coarse space problem s that the computaton of rght hand sde,.e. the nner product (φ 0, r) can be computed nsde Ω δ, therefore ts parallelzaton s straghtforward to perform and no communcatons are necessary. The restrcted addtve Schwarz method wth harmonc overlap (RASH) wth coarse space s then defne as P = P 0 + P. 5. Theoretcal Analyss. In order to analyze the method, we ntroduce a specal functon, denoted by ϑ whch ts energy sem-norm can be easly estmated and control the energy sem-norm of φ 0. The ϑ s used only for analyss purposes. We note that ϑ and φ 0 are dentcal ex cept at the nodes W δ,ovl. In next Lemma we study the energy sem-norm of each coarse bass functon. Lemma 5.. For any φ 0 and δgeq0 and ˆδ = O(δ). Then φ 0 2 H (Ω) = H C( + log((δ ) + h (2δ ) Proof. Consder frst the case n whch Ω δ s floatng square subdoman. Later we extend the cases n whch Ω δ s near the boundary. To smplfy our arguments, we assume n the analyss of ths and the remanng lemmas of the paper that the Ω δ and ts neghborng extended subdomans Ω δ j are squares of the same sze,.e. sdes length equals H + 2(δ. Ths assumpton s equvalent to that Ω R has sze H and δ level of overlap s appled. And also assume the overlap s not too large; for the analyss gven below (δ)h no larger than H/4 s enough. We note that too large overlap s not of practcal mportance. We note the our technques can be modfed to consder larger overlaps and more complex subdomans. We next partton Ω δ nto subregons n order to defne ϑ. ϑ s defned as a pecewse lnear functon of V wth support on Ω δ. It s equals to φ 0 on W \W,ovl δ. On the overlap regon W,ovl δ we defne ϑ carefully so that we can control ts energy sem-norm. Because φ 0 s dscrete harmonc on W,ovl δ, we have a(φ 0, φ 0) a(ϑ, ϑ ). Before we start defnng ϑ we need to ntroduce some notaton to characterze each pece of the subdoman Ω δ. We splt Ωδ nto subregons of four types: ΩI, Ωδδ, Ω δh, and Ω δ δ. The frst subregon, ΩI, s defned as Ω( W,non ). Ths subregon s therefore a square wth sdes of sze H 2δh. The second subregon, Ω δδ, s the place where Ω δ overlaps smulatneously three neghbors Ωδ j. Ωδδ therefore represents the 5

6 unon of the four corner peces of Ω δ,.e. four squares wth sdes of sze (2δ. The places where Ω δ overlaps only one neghbor are four rectangles wth sdes of szes H (2δ)h and (2δ. We then partton each of these four rectangles nto three subrectangles,.e. two of them are of Ω δ δ type and one of them of Ω δh type. For nstance, wthout lost of generalty assume we want the partton the ntersecton of Ω δ and and ts rght neghbor Ωδ j, excludng the the corner parts. Therefore regon we want to partton s a rectangle wth edges szes (2δ +)h n x drecton and H (2δ)h n y drecton. The partton of ths rectangles gves two subrectangles of Ω δ δ type wth dmensons 2(δ+)h (δ)h and whch each one has a whole edge n common wth a square of Ω δδ type. We denote them as transton subregons because t s placed between a corner type subregon Ω δδ and a face type subregon Ω δh. The Ω δh face type subregons are the subrectangles that are placed between the two subrectangles of Ω δ δ type. Ω δh face type regons have sdes of szes (2δ and H (4δ)h. On Ω I and we defne ϑ to be equals one,.e. equals to φ 0. Therefore φ 0 2 H (I) = ϑ 2 H (I) = 0. The most nterestng part s how to defne ϑ on Ω(W,ovl δ ). Bascally, Ω(W,ovl δ ) s the unon of corner, transton and face type regons defned above. We next defne ϑ n these regons and estmate ts energy sem-norm. The defnton of ϑ s gven by defnng ϑ (x) for all nodes x. We frst examne the case of a corner type square. Assume Q s such a square wth coordnates of ts vertces gven by V = [a, b], V 2 = [a + (2δ, b], V 3 = [a, b + (2δ ], and V 4 = [a + (2δ, b + (2δ ]. Assume also that V, V 2, and V 4 belong to Ω δ. In other words, the Q s placed on the southeast corner part of Ω δ. We then also ntroduce the square Q, wth vertces V 3 = [a, b + (2δ ], Ṽ = [a, b+(δ)h], Ṽ2 = [a+(δ +)h, b+(δ)h], and Ṽ4 = [a+(δ +)h, b+(δ +)h]. Note that Q s contaned n Q wth the vertex V 3 as the common vertex. We next defne ϑ on Q. We set ϑ (V 3 ) =, ϑ (Ṽ) = 0, ϑ (Ṽ2) = 0, ϑ (Ṽ4) = 0. The remanng nodes x on the edges ṼṼ2 and Ṽ2Ṽ4 we set ϑ (x) = 0, and on the edge V 3 Ṽ and V 3 Ṽ 4 we set ϑ (x) =. For nodes on Q\ Q we set ϑ (x) = 0. It remans only to defne ϑ (x) for nodes x n the nteror of Q. To defne ϑ there we use well-known cut functons technques such as those ntroduced n Lemma 4.4 of [5] but appled for two-dmensonal square regons. The most mportant observaton of the constructon of ϑ nsde Q s that ϑ (x) C/r near Ṽ or Ṽ4. Here r s dstance of x to Ṽ or Ṽ 4. Therefore, we obtan (see [5]) ϑ 2 H (Q) = ϑ 2 H ( Q) C( + log((δ )) h Snce nsde Ω δδ there four of those squares we obtan ϑ 2 H (Ω δδ ) C( + log((δ )) h We next consder the case of a transton type rectangle, denoted by T, whch we assume the coordnates of ts vertces are gven by V 3 = [a, b + (2δ ], V 4 = [a+(2δ+)h, b+(2δ+)h], V 5 = [a, b+(3δ+)h], and V 6 = [a+(2δ+)h, b+(3δ+)h]. Note that T stands on the top of the square Q ntroduced above and has the edge V 3 V 4 n common. We defne ϑ over the edge V 3 V 4 to be equals φ 0. Over the edge 6

7 V 3 V 5, we set ϑ =. Over edge V 4 V 6, we set ϑ = 0. And over edge V 5 V 6 we let ϑ to decreases lnearly from value to 0. It remans to defne ϑ nsde T. Let us defne the nodes V l = [a + (δ)h, b + (2δ ] and V r = [a + (δ, b + (2δ ]. These nodes are exactly the places over the sde V 3 V 4 where φ 0 jumps from to 0. On the trangle V 3 V l V 5 we set ϑ =. On the trangle V r V 4 V 6 we set ϑ = 0. On the regon V l V r V 6 V 5, we defne ϑ to decreases lnearly n the x drecton from value to 0. We note that next to the nodes V l V r, ϑ has a sngular behavor smlar to ϑ (x) C/r where r s the dstance from x to the lne V l V r. Agan, we also obtan ϑ 2 H (T ) C( + log((δ )). h Snce nsde of Ω δ δ contans eght rectangles of T type we obtan ϑ 2 H (Ω δ δ C( + log((δ )). ) h Fnally we consder the rectangle of face type, denoted by R, assumng that the coordnates of ts vertces are gven by V 5 = [a, b + (3δ ], V 6 = [a + (2δ, b + (3δ ], V 7 = [a, b + H (δ )h], and V 8 = [a + (2δ, b + H (δ )h]. Note that R s on the top of the rectangle T defned above and ts heght s H (4δ)h. The vertces V 6 and V 8 are the vertces that belong to Ω δ. We defne ϑ (x) = f x s on the edge V 5 V 7, and equals zero f x n on the edge V 6 V 8, and lnear n x drecton on the remanng ponts. We obtan then ϑ 2 H (R) H (4δ)h (2δ. Snce there are four of those rectangles nsde Ω δh, we obtan ϑ H (Ω δh ) C H (4δ)h (2δ C H (2δ. Puttng all every peces ϑ together, we can see that ϑ V and t s equals to φ 0 on W \W δ,non. The lemma follows by usng that φ 0 at the φ 0 H (Ω δ ) ϑ H (Ω δ ) C( + log( (δ H ) + h (2δ ) by usng all the estmates of subregons of four types dscussed. For the cases n whch Ω s near the boundary Ω the analyss does not change much. The analyss s smlar and we deal stll wth the four types subregons dscussed above. The basc dfference s that the sze of the T, R, I, and Q change slghtly from before, however stll at the same order of magntude ˆδ = O(δ). And the proof follows. Lemma 5.2. For any u V we have I 0 u 2 H (Ω) H C( + log((δ ) + h (2δ ) u 2 H (Ω). Proof. Let I 0 u = N = ūφ 0, where ū = Ω δ Ω δ udx. 7

8 Let c be a constant. By usng Cauchy-Schwarz nequalty we have ū c = Ω δ (u c)dx C H u c L 2 (Ω δ ). Ω δ Assume that Ω δ j s neghbor to Ωδ,.e. Ωδ j overlaps Ωδ j. Usng a trangular nequalty we obtan ū ū j ū c + ū j c C H u c L 2 (Ω δ Ωδ j ). And usng a Bramble-Hlbert argument we obtan (5.) ū ū j C nf c H u c L 2 (Ω δ Ωδ j ) C nf c H u c L 2 (Ω δ,ext ) C u H (Ω δ,ext ). Here Ω δ,ext s the unon of Ω δ and ts all overlappng neghbors extended regons Ωδ k. Note that we had used that Ω δ,ext has a regular shape and has sze O(H) to use a Fredrch nequalty. We now are ready to evaluate the energy norm of I 0 u on Ω δ. We use strongly the fact that φ 0 =, on Ω δ. j= On Ω I, φj 0 vanshes when j and equals one when j =. Therefore, I 0 u 2 H (I) = ū 2 H (I) = 0. We next evaluate I 0 u on a corner type square Q. Wthout less of generalty, ths corner s the southeast corner of Ω δ, and overlaps wth Ωδ, Ω δ 2, and Ω δ 3,.e. the east, south, and southeast regons. Then We next use that to obtan and by a trangular nequalty I 0 u 2 H (Q) = ū φ 0 + φ j= 3 j= φ k 0 = on Q ū k φ k 0 2 H (Q). 3 I 0 u 2 H (Q) = (ū k ū )φ k 0 2 H (Q), k= k= 3 3 I 0 u 2 H (Q) C (ū k ū )φ k 0 2 H (Q) C ū k ū φ k 0 2 H (Q). 8 k=

9 Usng (5.) and same arguments n the proof of Lemma 5. we obtan I 0 u 2 H (Q) C( + log((δ )) u h H (Ω δ,ext ). Usng smlar arguments, we obtan for a rectangle of face type R and transton type T, I 0 u 2 H (R) C( + H (2δ ) u 2 H (Ω δ,ext ) I 0 u 2 H (T ) Summng all the contrbutons, we have I 0 u 2 H (Ω δ C( + log((δ )) u 2 h. H (Ω δ,ext ) H ) C( + log((δ ) + h (2δ ) u 2. H (Ω δ,ext ) Agan, the analyss s smlar for the cases n whch Ω s near the boundary. The bounds wll be of the same order as before f ˆδ = O(δ). And the proof follows. Theorem 5.. There exsts a constant C > 0, ndependent of h and the number of subdomans, such that for any u Ṽ, there exst u Ṽ, such that and u = u, =0 (5.2) u 2 H (Ω) =0 C[( + log((δ ))( + log( H h h )) + H (2δ ] u 2 H (Ω) Recall that δ s a nonnegatve nteger ndcatng the number of overlap and δh s the actual overlappng sze. Proof. Let the local functons v Ṽ be defned as { u(xk ) f f node x k W,n δ v (x k ) = 0 f f node x k W \ W δ and v s dscrete harmonc at the x k W δ,ovl Let u 0 Ṽ0 be defned as u 0 = I 0 u = I0u = ū φ 0. = = The decomposton u = N =0 u s defned as follows u = v ū φ 0, =,..., N 9

10 u 0 = ū φ 0. = The next step s to bound N =0 u 2 H (Ω). For boundng u 0 2 )H (Ω) we use Lemma 5.2,.e u 0 2 H (Ω) H C( + log((δ ) + h (2δ ) u 2 H (Ω). To bound u 2 H (Ω), =,..., N, we use the ϑ, =,..., N ntroduced n Lemma 5.. We ntroduce ũ Ṽ as follows ũ (x) = I h (ϑ (x)(u(x) ū )). Note that ũ (x) s equal to u (x) n the whole Ω but Ω(W δ,ovl ). On Ω(W δ,ovl ) u s dscrete harmonc. Therefore, we have u 2 H (Ω) ũ 2 H (Ω). So, the next step s to bound ũ 2 H (Ω). Let K be an element of Ωδ denote w = u ū. We obtan ũ 2 H (K) = I h(ϑ w ) 2 H (K) 2 ϑ w 2 H (K) + 2 I h(( ϑ ϑ )w ) 2 H (K). and let us Here, ϑ s the average of ϑ on K, and I h s the standard pontwse nterpolator. To bound the frst part we use that ϑ, to obtan ϑ w 2 H (K) ϑ (u ū ) 2 H (K) u ū 2 H (K) C u 2 H (K). The last nequalty comes from the fact that ū s a constant. For the second part we frst use an nverse nequalty to have I h (( ϑ ϑ )w ) 2 H (K) C h 2 I h(( ϑ ϑ )w ) 2 L 2 (K). We next splt the regon Ω δ n the 4 peces that we had already ntroduced n Lemma 5.,.e. Ω I, ΩδH, Ω δ δ and Ω δδ. For K on Ω I, ϑ ϑ s dentcally to zero, and therefore, I h (( ϑ ϑ )w ) 2 L 2 (K) vanshes. For K on Ω δh we use that ϑ ϑ 2 L (K) C( h (2δ )2 to obtan h 2 I h(( ϑ ϑ )w ) 2 L 2 (K) C ((2δ ) 2 w 2 L 2 (K). And we use small overlap technques Dryja and Wdlund [6] to obtan ((2δ ) 2 w 2 L 2 (Ω H δ ) C( + H (2δ w 2 H (Ω δ ) + H((2δ ) w 2 L 2 (Ω δ )) 0

11 We use that w 2 H (Ω δ ) = u 2 and a Fredrch nequalty H (Ω δ ) Puttng everythng together, we obtan K Ω δδ w 2 L 2 (Ω δ ) CH2 u 2 H (Ω δ ). (2δ ) 2 w 2 L 2 (Ω H δ ) C( + H (2δ ) u 2 H (Ω δ ) For K on Ω δδ, we use smlar arguments as n Dryja, Smth and Wdlund [5] to obtan ((δ ) 2 I h(( ϑ ϑ )w 2 L 2 (K) r 2 w 2 L 2 (K) K Ω δδ K Ω δδ where ch r C((δ ) s the dstance to those cut peces. We have used here that ϑ has the sngular behavor C/r on Ω δδ. We have then C(δ+)h r 2 w 2 L 2 (K) C r 2 r w 2 L (Ω δδ)dθdr ch θ C( + log( (δ h )) w 2 L (Ω δδ ) C[( + log((δ ))( + log( H h h ))] u 2 Ω. δ For the last nequalty we had used a well-known result (see Bramble []) u ū 2 L (Ω δδ ) u ū L (Ω δ ) C(+log( H h )) u ū 2 H (Ω δ )andusngthat ū s the average of u on Ω δ,.e. a Fredrch nequalty u ū 2 H (Ω δ ) C( + log( H h )) u 2 H (Ω δ ) Usng smlar arguments, we also obtan a smlar bound fot the Ω δ δ. Agan, the analyss s smlar for the cases n whch Ω s near the boundary. The bounds wll be of the same order as before f ˆδ = O(δ). And the proof follows. 6. Numercal experments. In ths secton, we present some numercal results. We consder the Posson s equaton on the unt square wth zero Drchlet boundary condton. We consder the PCG, wth RASH or AS as precondtoner. In order to apply the RASH/PCG method, t s necessary to force the soluton to belong to Ṽ. To do so, an addtonal pre computaton s needed to reduce to ths case. Ths s done by applyng one precondtoned Rchardson teraton gven by w = ( R δ ) T (Ãδ ) R0 f. = We note that u = u w Ṽ, and therefore we can apply the RASH/PCG. The AS/CG s the classcal addtve Schwarz precondtoned CG as descrbed n [?]. We shall use a fne mesh. The stoppng condton for CG s to reduce the ntal

12 Table 6. RASH (AS) performance results wthout coarse solver for solvng the Posson s equaton on a mesh decomposed nto 2 2 = 4 subdomans wth overlap = ovlp. ovlp Iter Cond Max Mn 0 42 (42) 29.(29.).98 (.98) (0.054) 24+ (28) 48.4 (86.3).94 (4.00) (0.0464) (23) 33.3 (5.8).9 (4.00) (0.0773) 3 8+ (20) 27.2 (37.0).89 (4.00) (0.08) Table 6.2 RASH (AS) performance results wthout coarse solver for solvng the Posson s equaton on a 32 DOM 32 DOM mesh decomposed nto DOM DOM subdomans wth overlap =. DOM DOM Iter Cond Max Mn (20) 26.8 (43.7).89 (4.00) (0.096) (42) 86.9 (45.).95 (4.00) (0.0276) (78) 328. (550.).97 (4.00) (0.0073) (56) 295 (268.).98 (4.00) 0.005(0.008) resdual by a factor 0 6. The teraton counts, condton numbers, maxmum and mnmum egenvalues are summerzed n Tables 6, 6, and 6. From Tables 6 and 6, t s clear that RASH/PCG s always better than the classcal AS/PCG n terms of the teraton counts and condton numbers for the case that no coarse space s used. Ths s an mportant result snce t s easy to modfy a (parallel) code wth AS/PCG to RASH/PCG. We also predct that the RASH/CG would be even better than AS/CG n a parallel computer wth dstrbuted memory snce much less communcatons are requred. Also the local solvers are slghtly cheaper snce the local solvers has a slghtly smaller number of unknowns than for the regular AS. We also tested our new coarse space for the RASH/PCG. Table 6 shows that ths coarse space s very effectve when the number of subdomans gets very large. Ths s encouragng and we predct that ths coarse can be very promsng for parallel mplementaton snce t s easy to program, t requres fewer communcatons and computatons than the exstng coarse spaces. The results are for the case we use u(x, y) = e 5(x+y) sn(πx) sn(πy) as the exact soluton. We note that dfferent rght hand sdes and soluton functons were tested; the results follow the same patterns as below. REFERENCES [] J. Bramble, A second order fnte dfference analogue of the frst bharmonc boundary value problem, Numer. Math., 9, 966, pp [2] X.-C. Ca e M. Sarks, A restrcted addtve Schwarz precondtoner for general sparse lnear systems, SIAM J. Sc. Comput., 2 (999), pp [3] M. Dryja and O. Wdlund, An addtve varant of the Schwarz alternatng method for the case of many subregons, Techncal Report 339, also Ultracomputer Note 3, Department of Computer Scence, Courant Insttute, 987 [4] M. Dryja, M. Sarks, and O. Wdlund, Multlevel Schwarz methods for ellptc problems wth dscontnuous coeffcents n three dmensons, Numer. Math., Vol. 72, 996, pp [5] M. Dryja, B. Smth, and O. Wdlund, Schwarz analyss of teratve substructurng algo- 2

13 Table 6.3 RASH performance results wth a coarse solver for solvng the Posson s equaton on a 32 DOM 32 DOM mesh decomposed nto DOM DOM subdomans wth overlap = (2). DOM DOM Iter Cond Max Mn (6+) 30.2 (22.3) 2.67 (2.63) (0.79) (33+) 46.6 (32.6) 2.9 (2.88) (0.0886) (44+) 57.7 (4.0) 2.96 (2.93) (0.077) (46+) 6.4 (44.0) 2.96 (2.94) (0.0669) rthms for ellptc problems n three dmensons, SIAM J. Numer. Anal., Vol. 3(6), 994, pp [6] M. Dryja and O. Wdlund, Doman decomposton algorthms wth small overlap, SIAM J. Sc. Comp., 5, 994, pp [7] M. Dryja and O. Wdlund, Schwarz Methods of Neumann-Neumann type for threedmensonal ellptc fnte elements problems, Comm. Pure Appl. Math, Vol. 48, 995, pp [8] C. Farhat and F. Roux, A Method of Fnte Element Tearng and Interconnectng and ts Parallel Soluton Algorthm, Int. J. Numer. Mech. Engrg., Vol. 32, 99, pp [9] J. Mandel, Balancng Doman Decomposton, Comm. Numer. Meth. Eng., Vol. 9, 993, pp [0] A. Matsokn and S. Nepomnyaschkh, A Schwarz alternatng method n a subspace, Sovet Mathematcs, Vol. 29(0), 985, pp [] M. Sarks, Nonstandard coarse spaces and Schwarz methods for ellptc problems wth dscontnuous coeffcents usng non-conformng elements, Numersche Mathematk, Vol 77, 997, [2] O. Wdlund, Exotc Coarse Spaces for Schwarz Methods for Lower Order and Spectral Fnte Element, Contemporary Mathematcs, Seventh Internatonal Conference of Doman Decomposton Methods n Scentfc and Engneerng Computng, Davd E. Keyes and Jnchao Xu, Vol 80, 994,