Additive Schwarz Method for DG Discretization of Anisotropic Elliptic Problems

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1 Addtve Schwarz Method for DG Dscretzaton of Ansotropc Ellptc Problems Maksymlan Dryja 1, Potr Krzyżanowsk 1, and Marcus Sarks 2 1 Introducton In the paper we consder a second order ellptc problem wth dscontnuous ansotropc coeffcents defned on a polygonal regon Ω. The problem s dscretzed by a Dscontnuous Galerkn (DG) fnte element method wth trangular elements and pecewse lnear functons. Our goal s to desgn and analyze an addtve Schwarz method (ASM), see the book by Tosell and Wdlund [4], for solvng the resultng dscrete problem wth rate of convergence ndependent of the jumps of the coeffcents. The method s two-level and wthout overlap of Ω l, the substructures nto whch the orgnal regon Ω s parttoned. It s proved that the convergence of the method s ndependent of the jumps of the coeffcents appearng on trangles nsde of Ω l, see [3]. It s the same for the jumps appearng on trangles whch touch Ω l under addtonal assumptons on the coeffcents, lke monotoncty or quasmonotoncty. The ASM dscussed here s a generalzaton of method presented n [1]. Numercal experments confrm the theoretcal results. The paper s organzed as follows. In Secton 2, dfferental and dscrete DG problems are formulated. In Secton 3, ASM for solvng the dscrete problem s desgned and analyzed. Numercal experments are presented n Secton 4. 2 Dfferental and dscrete DG problems We consder the followng ellptc problem: fnd u H0 1 (Ω) such that a(u,v) = f (v), v H 1 0 (Ω) (1) where a(u,v) = ρ(x) u vdx, f (v) = Ω ( ) ρ11 (x) ρ ρ(x) = 12 (x). ρ 21 (x) ρ 22 (x) Ω f vdx, 1 Unversty of Warsaw, Poland; e-mal: {m.dryja,p.krzyzanowsk}@mmuw.edu.pl 2 Worcester Polytechnc Insttute, USA and Insttuto Naconal de Matemátca Pura e Aplcada, Brasl; e-mal: msarks@wp.edu 1

2 2 Maksymlan Dryja, Potr Krzyżanowsk, and Marcus Sarks We assume that Ω s a polygonal regon, f L 2 (Ω) and ρ(x), the dffusvty tensor, s a symmetrc matrx, unformly postve defnte wth respect to x, and ρ j L (Ω),. j = 1,2. Under these assumptons problem (1) s well posed. Let T h (Ω) be a trangulaton of Ω wth trangular elements K and the mesh parameter h. We assume that T h (Ω) s shape regular and quasunform. Let X (K ) denote a space of lnear functons on K and N X h (Ω) = Π=1 N X (K ), Ω = K =1 be the space n whch problem (1) s approxmated. Note that X h (Ω) H 1 (Ω) and ts elements do not vansh on Ω, n general. The dscrete problem for (1) s of the form: fnd u h X h(ω) such that â h (u h,v h) = f (v h ), v h X h (Ω), (2) where for u,v X h (Ω),u = {u } N =1,u X (K ), and and ρ () kl â h (u,v) = N =1 ρ () = ρ K, â (u,v), f (v) = N =1 ρ () = {ρ () kl }2 k,l=1, K f v dx are constants on K whch can always be assumed for lnear elements. Here wth symmetrc forms s (u,v) = E j K â (u,v) = a (u,v) + s (u,v) + p (u,v), p (u,v) = a (u,v) = ρ () u v dx, K E j ω [n T ρ () u (v j v ) + n T ρ () v (u j u )]ds, σ h E j K E j γ j (u u j )(v v j )ds where E j = E j = K K j,e j K and E j K j ; n = n E j s the unt normal vector to E j pontng from K to K j ; and ω ω E j = δ ( j) ρ n δ () ρ n + δ ( j) ρ n, ω j ω E j = δ () ρ n δ () j) ρn + ρ( ρ n

3 Addtve Schwarz Method for DG Ansotropc Ellptc Problems 3 δ () ρ n = nt ρ () n, δ ( j) ρ n = nt j ρ ( j) n j ; γ j γ E j = 2δ () ρ n δ ( j) ( j) ρn /(δ ρ n + δ () ρn ); σ s a postve (suffcently large, cf. Lemma 1) penalty parameter, whch ensures the ellptcty of â (, ). To analyze problem (2) we ntroduce some auxlary blnear forms and a broken norm. Let the ellptc symmetrc form d h (, ) be defned as d h (u,v) = N =1 d (u,v), d (u,v) = a (u,v) + p (u,v) (3) and let the weghted broken norm n X h (Ω) be defned by u 2 1,h d h(u,u) = N =1 { (ρ () ) 1/2 u 2 L 2 (K ) + σ h γ j u u j 2 L 2 (E }. j) E j K Lemma 1. There exsts σ 0 > 0 such that for σ σ 0 there exst postve constants C 0 and C 1 ndependent of ρ () and h such tha C 0 d (u,u) â (u,u) C 1 d (u,u) (4) and for all u X h. C 0 d h (u,u) â(u,u) C 1 d h (u,u) For the proof we refer for example to [1] for sotropc cases and [2] for ansotropc cases. Lemma 1 mples that the dscrete problem (2) s well posed f the penalty parameter σ σ 0. Below σ s fxed and assumed to satsfy the above condton. The error bound s gven by Theorem 1. Let u and u h be the solutons of (1) and (2). For u K H 2 (K ) holds u u N h 2 1,h Mh2 λ max (ρ () ) u 2 H 2 (K ) =1 where M s ndependent of h,u and ρ ; λ max (ρ () ) s a maxmum egenvalue of ρ (). The proof follows from Lemma 1, for detals see for example [2]. 3 Addtve Schwarz method We desgn and analyze ASM for solvng problem (2) followng to the abstract theory of ASMs, see for example, [4].

4 4 Maksymlan Dryja, Potr Krzyżanowsk, and Marcus Sarks 3.1 Decomposton of X h (Ω) Let Ω = L l=1 Ω l, Ω l Ω m = {/0}, l m where Ω l s a unon of trangulaton elements K and H l = dam(ω l ). The decomposton of X h (Ω) s where for l = 1,...,L and for l = 0 X h (Ω) = X (0) (Ω) + X (1) (Ω) X (L) (Ω), X (l) (Ω) = {v = {v } N =1 X h(ω) : v = 0 on K Ω l } V (0) (Ω) = span{φ (l) } L l=1 wth φ (l) = 1 on Ω l and φ (l) = 0 otherwse. 3.2 Inexact local solvers For u (l) = {u (l) } N =1 X (l) (Ω) and v (l) = {v (l) } N =1 X (l) (Ω),l = 1,...,L, we defne b l (u (l),v (l) ) = d h (u (l),v (l) ). The overlap between local subproblems s very small (only through the subdoman nterface), reducng communcaton cost to a level smlar to substructurng methods. Instead of solvng exact subproblems wth form â h (, ) on subdomans, we solve problems wth smplfed form d h (, ). Note that on X (l) (Ω) X (l) (Ω) d h (u (l),v (l) ) = {(ρ () u (l), v (l) ) L 2 (K ) + σ K Ω l E j K h γ j(u (l) u (l) j,v (l) v (l) j ) L 2 (E j ) }. For l = 0 and u (0) = {u (0) } N =1 X (0) (Ω) and v (0) = {v (0) } N =1 X (0) (Ω) we set b 0 (u (0),v (0) ) = d h (u (0),v (0) ) L l=1 σ h γ j (u (0) u (0) j,v (0) v (0) j ) L 2 (E j ). E j Ω l 3.3 Operator equaton For l = 0,...,L, let us defne T l : X h (Ω) X (l) (Ω) by

5 Addtve Schwarz Method for DG Ansotropc Ellptc Problems 5 b l (T l u,v) = â h (u,v), v X (l) (Ω). Then problem (2) s replaced by Tu h = g h, g h = L l=0 g l, g l = T l u h. (5) wth T = T 0 + T T L. Note that n order to compute g l we do not need to know u h. From the theorem below t follows that problems (2) and (5) have the same unque soluton. 3.4 Analyss Let Ω l h denote a layer around Ω l. It s a unon of K Ω l whch touch Ω l by edge or/and vertex. Let ᾱ l := max λ max (ρ () ), α l := mn λ mn (ρ () ) K Ω l h K Ω l h where λ max (ρ () ) and λ mn (ρ () ) are maxmum and mnmum egenvalues of ρ () on K. Theorem 2 (man result). For any u X h (Ω) there holds where C 2 β 1 â h (u,u) â h (Tu,u) C 3 â h (u,u) (6) ᾱ l Hl 2 β = max 1 l L α l h 2 and C 2 and C 3 are postve constants ndependent of ρ (),ᾱ l and α l for = 1,...,N and l = 1,...,L. To prove Theorem 2 we need to check three key assumptons of the abstract theory of ASMs, see Tosell and Wdlund book [4]. The proof s omtted here due to the lmt of pages and wll be publshed elsewhere. Remark 1. Note that the convergence of the method s ndependent of the jumps of ρ () on Ω l \Ω h l for all l = 1,...,L,.e. of the jumps of ρ () on K whch do not touch Ω l. Remark 2. Let us menton several specfc cases when the above estmate can be mproved. When ρ s sotropc and subdomanwse constant, then we can prove that β = max l (H l /h) n (6). When ᾱ l and α l are the same order and α l max K Ω l λ mn (ρ () ),

6 6 Maksymlan Dryja, Potr Krzyżanowsk, and Marcus Sarks then β = max l (H l /h),.e. the convergence s ndependent of the jumps of ρ (). Estmate (6) can be also mproved n the case when λ max (ρ () ) on K whch touch Ω l by edges are monotonc or quas-monotonc on Ω l for l = 1,...,L. 4 Numercal experments Let us choose the unt square as the doman Ω and for some prescrbed nteger m dvde t nto L = 2 m 2 m smaller squares Ω l (l = 1,...,L) of equal sze. Ths decomposton of Ω s then further refned nto a unform trangulaton T h (Ω) based on a square 2 M 2 M grd (M m) wth each square splt nto two trangles of dentcal shape. Hence, the fne mesh parameter h = 2 M, whle the coarse grd parameter s H = 2 m. We dscretze system (1) on the fne trangulaton usng method (2) wth σ = 7. In tables below we report the number of Precondtoned Conjugate Gradent teratons for operator T (defned n Secton 3.3) whch are requred to reduce the ntal Eucldean norm of the resdual by a factor of 10 6 and (n parentheses) the condton number estmate for T. We consder two sets of test problems: wth ether ansotropc or dscontnuous coeffcents matrx ρ. We wll always choose a random vector for the rght hand sde and a zero as the ntal guess. Dscontnuous, elementwse constant sotropc coeffcents. Let us consder dffuson coeffcent of the form ρ(x) = ρ 11 (x) I (7) where ρ 11 equals 1 on even numbered elements (of fne trangulaton) and equals 10 2 on odd ones. Table 1 shows the dependence on the rato between H and h n ths case. Fne (M) Coarse (m) 2 33 (32) 82 (300) 133 (530) 164 (840) 237 (2000) 3 45 (41) 140 (370) 189 (700) 225 (1100) 4 48 (42) 155 (470) 186 (690) 5 41 (48) 155 (470) 6 49 (44) Table 1 Dependence of the number of teratons and the condton number (n parentheses) on the rato H/h, where H = 2 m and h = 2 M. Isotropc, elementwse constant coeffcent. Next, let us fx the number of subdomans and the fne mesh sze so that M = 3 and m = 5 and thus H/h = 4. Table 2 shows the dependence of the convergence rate and the condton number as we vary the value of ρ 11 on odd-numbered trangles; on even trangles t remans equal to 1 as prevously.

7 Addtve Schwarz Method for DG Ansotropc Ellptc Problems 7 ρ ter (cond) 61 (80) 72 (10 2 ) 167 ( ) 335 ( ) 485 ( ) 613 ( ) 743 ( ) Table 2 Dependence of the number of teratons and the condton number (n parentheses) on the dscontnuty n the sotropc, elementwse constant coeffcent. Fxed H/h = 4. Indeed, the condton number estmates agree well wth our theory regardng the dependence on the dscontnuty of the coeffcent. In our testcase the ncrease n the condton number s rather lnear than quadratc n H/h, as reported n Table 1. Ths behavour s n agreement wth our Remark 2. Let us also explan that low teraton numbers n Table 2 are due to a very rapd resdual n the resdual durng the ntal phase of the teraton. Dscontnuous, domanwse constant sotropc coeffcents. Here we consder ρ as n (7), wth dscontnutes algned wth an auxlary parttonng of Ω nto 4 4 squares. Precsely, we ntroduce a red black checkerboard colorng of ths parttonng and set ρ = 1 n red regons, and the value of ρ 11 reported n Table 3 n black ones. In ths way, our decomposton of the doman wth M = 5 and m = 3 wll always be algned wth the dscontnutes and Table 3 shows the dependence on ρ 11 n ths case. ρ ter (cond) 61 (80) 60 (70) 58 (67) 58 (68) 62 (68) 64 (68) 67 (68) Table 3 Dependence of the number of teratons and the condton number (n parentheses) on the dscontnuty when the coeffcent s sotropc and constant nsde subdomans. Red black 4 4 dstrbuton of ρ, algned wth doman decomposton. Fxed H/h = 4. As predcted n Remark 2, there s no dependence on the dscontnuty n the coeffcents n ths case untl the coeffcent remans contnuous (constant) nsde subdoman. Ths behavour s not observed when the red black parttonng s not algned wth the subdomans Ω l : correspondng numbers for a 3 3 parttonng are shown n Table 4. ρ ter (cond) 62 (80) 68 (130) 85 (710) 96 ( ) 113 ( ) 126 ( ) 140 ( ) Table 4 Dependence of the number of teratons and the condton number (n parentheses) on the dscontnuty when the coeffcent s sotropc and dscontnuous across subdoman boundares. Red black 3 3 dstrbuton of ρ, not algned wth the doman decomposton. Fxed H/h = 4. Ansotropc, dscontnuous coeffcents. Let us contnue wth the 4 4 red black parttonng and let us set the coeffcent matrx ρ equal to ρ R n red regons and ρ B n black ones, where ρ R (x) = ( 10 + ρ22 0 ) 0 ρ 22, ( ) ρ B ρ22 0 (x) =, ρ 22

8 8 Maksymlan Dryja, Potr Krzyżanowsk, and Marcus Sarks wth constant ρ 22 as specfed n Table 5. In ths way ρ s constant n both red and black regons, but t suffers from dscontnuty across the parttonng borders; the jump s always equal to 10, whle the ansotropy rato s /ρ 22. The condton numbers grow lnearly wth the growth of ρ 22, whch agrees wth Theorem 2. ρ ter (cond) 60 (82) 94 (210) 222 (10 3 ) 463 (10 4 ) 680 (10 5 ) 782 (10 6 ) 897 (10 7 ) Table 5 Dependence on the ansotropy for dscontnuous, pecewse constant coeffcent. Fxed H/h = 4. Ansotropc, constant coeffcents. Fnally, let us consder ( ) 1 0 ρ(x) = 0 ρ 22 wth ρ 22 constant throughout entre Ω, assumng values specfed n Table 6. ρ ter (cond) 60 (82) 74 (10 2 ) 159 ( ) 159 ( ) 144 ( ) 143 ( ) 124 ( ) Table 6 Dependence on the ansotropy. Fxed H/h = 4. Contnuous, constant coeffcent. It turns out that after ntal lnear ncrease n the condton number for moderate ρ 22, the condton number s nsenstve to further growth of the ansotropy rato ρ 22. Ths observaton can also be explaned on the ground of our theory; the detals wll be provded elsewhere. Acknowledgements The research of the frst two authors has been partally supported by the Polsh Natonal Scence Centre grant 2011/01/B/ST1/ References 1. Dryja, M., Sarks, M.: Addtve average Schwarz methods for dscretzaton of ellptc problems wth hghly dscontnuous coeffcents. Comput. Methods Appl. Math. 10(2), (2010) 2. Ern, A., Stephansen, A.F., Zunno, P.: A dscontnuous Galerkn method wth weghted averages for advecton-dffuson equatons wth locally small and ansotropc dffusvty. IMA J. Numer. Anal. 29(2), (2009). DOI /manum/drm050. URL /manum/drm Graham, I.G., Lechner, P.O., Schechl, R.: Doman decomposton for multscale PDEs. Numer. Math. 106(4), (2007). DOI /s URL org/ /s Tosell, A., Wdlund, O.: Doman decomposton methods algorthms and theory, Sprnger Seres n Computatonal Mathematcs, vol. 34. Sprnger-Verlag, Berln (2005)