How To Solve A 3D Problems In A 3D Problem With A 3Rd Dimension

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1 Some Recent Tools and a BDDC Algorthm for 3D Problems n H(curl) Clark R. Dohrmann 1 and Olof B. Wdlund 2 1 Sanda Natonal Laboratores, Albuquerque, New Mexco, , USA. Sanda s a multprogram laboratory operated by Sanda Corporaton, a Lockheed Martn Company, for the Unted States Department of Energy s Natonal Nuclear Securty Admnstraton under contract DE-AC04-94AL85000, crdohrm@sanda.gov 2 Courant Insttute, 251 Mercer Street, New York, NY 10012, USA. Ths work supported n part by the U.S. Department of Energy under contracts DE-FG02-06ER25718 and n part by Natonal Scence Foundaton Grant DMS , wdlund@cms.nyu.edu, Summary. We present some recent doman decomposton tools and a BDDC algorthm for 3D problems n the space H(curl;Ω). Of prmary nterest s a face decomposton lemma whch allows us to obtan mproved estmates for a BDDC algorthm under less restrctve assumptons than have appeared prevously n the lterature. Numercal results are also presented to confrm the theory and to provde addtonal nsghts. 1 Introducton We nvestgate a BDDC algorthm for three-dmensonal (3D) problems n the space H 0 (curl;ω). The subject problem s to obtan edge fnte element approxmatons of the varatonal problem: Fnd u H 0 (curl;ω) such that a Ω (u, v) = ( f, v) Ω v H 0 (curl;ω), where a Ω (u, v) := [(α u v) + (β u v)]dx, ( f, v) Ω = f vdx. Ω Ω The norm of u H(curl;Ω), for a doman wth dameter 1, s gven by a Ω (u, u) 1/2 wth α = 1 and β = 1; the elements of H 0 (curl) have vanshng tangental components on Ω. We could equally well consder cases where ths boundary condton s mposed only on one or several subdoman faces whch form part of Ω. We wll assume that α 0 and β > 0 are constant n each of the subdomans Ω 1,...,Ω N. Our results could be presented n a form whch accommodates propertes whch are not constant or sotropc n each subdoman, but we avod ths generalzaton for purposes of clarty. In the poneerng work of [11], two dfferent cases were analyzed for FETI-DP algorthms: Case 1: α = α for = 1,...,N

2 16 Clark R. Dohrmann and Olof B. Wdlund The condton number bound reported for the precondtoned operator s where H/h := max H /h. Case 2: κ C max (1 + H 2 β /α)(1 + log(h/h)) 4, (1) β = β for = 1,...,N for whch the reported condton number bound s κ C max (1 + H 2 β/α )(1 + log(h/h)) 4. (2) We address the followng basc questons regardng [11] n ths study. 1. Is s possble to remove the assumpton of α = α or β = β for all? 2. Is t possble to remove the factor of H 2 β /α from the estmates? 3. Is s possble to reduce the logarthmc factor from four powers to two powers as s typcal of other teratve substructurng algorthms? 4. Do FETI-DP or BDDC algorthms for 3D H(curl) problems have certan complcatons not present for problems wth just a sngle parameter? We fnd n the followng sectons that the answers are yes to all four questons. However, due to page lmtatons, we only consder here the relatvely rch coarse space of Algorthm C of [11]. We remark that the analyss of 3D H(curl) problems wth materal property jumps between subdomans s qute lmted n the lterature. A comprehensve treatment of problems n 2D can be found n [3]. A dfferent teratve substructurng algorthm for 3D problems s gven n [6], but the authors were unable to conclude whether ther condton number bound was ndependent of materal property jumps. 2 Tools We assume that Ω s decomposed nto N non-overlappng subdomans, Ω 1,...,Ω N, each the unon of elements of the trangulaton of Ω. We denote by H the dameter of Ω. The nterface of the doman decomposton s gven by ( N Γ := Ω )\ Ω, =1 and the contrbuton to Γ from Ω by Γ := Ω \ Ω. These sets are unons of subdoman faces, edges, and vertces. For smplcty, we assume that each subdoman s a shape-regular and convex tetrahedron or hexahedron wth planar faces. We assume a shape-regular trangulaton T h of each Ω wth nodes matchng across the nterfaces. The smallest element dameter of T h s denoted by h. Assocated wth the trangulaton T h are the two fnte element spaces W h grad H(grad,Ω ) and W h curl H(curl,Ω ) based on contnuous, pecewse lnear, tetrahedral nodal elements and lnear, tetrahedral edge (Nédeléc) elements, respectvely. We could equally well develop our algorthms and theory for low order hexahedral elements. The energy of a vector functon u W h curl for subdoman Ω s defned as E (u) := α ( u, u) Ω + β (u, u) Ω, (3)

3 Some Recent Tools and a BDDC Algorthm for 3D Problems n H(curl) 17 where α and β are assumed constant n Ω. Let N e W h curl and t e denote the fnte element shape functon and unt tangent vector, respectvely, for an edge e of T h. We assume that N e s scaled such that N e t e = 1 along e. The edge fnte element nterpolant of a suffcently smooth vector functon u H(curl,Ω ) s then defned as Π h (u) := u e N e, u e := (1/ e ) u t e ds, (4) e M e Ω where M Ω s the set of edges of T h, and e s the length of e. We wll also make use of other sets of subdoman edges. The sets M Ω, M E, M F, and M F contan the edges of Ω, subdoman edge E, subdoman face F, and F, respectvely. We denote by G F, G E, and G V sets of subdoman faces, subdoman edges, and subdoman vertces for Ω. The wre basket W s the unon of all subdoman edges and vertces for Ω. We wll also make use of the symbol ω := 1 + log(h /h ), and bold faced symbols refer to vector functons. We denote by p the mean of p over Ω. The estmate n the next lemma can be found n several references, see e.g., Lemma 4.16 of [12]. Lemma 1. For any p W h grad and subdoman edge E of Ω, p 2 L 2 (E ) Cω p 2 H 1 (Ω ). (5) Lemma 2. For any p W h grad, there exst p V, p E, p F W h grad such that p Ω = p V Ω + p E Ω + p F Ω, (6) V G V E G E F G F where the nodal values of p V, p E, and p F on Ω may be nonzero only at the nodes of V, E, and F, respectvely. Further, p V 2 H 1 (Ω ) C p 2 H 1 (Ω ), (7) p E 2 H 1 (Ω ) Cω p 2 H 1 (Ω ), (8) p F 2 H 1 (Ω ) Cω2 p 2 H 1 (Ω ). (9) Proof. The estmates n (7-9) are standard, and follow from Corollary 4.20 and Lemma 4.24 of [12] and elementary estmates. We note that a Poncaré nequalty allows us to replace the H 1 -norm of p by ts H 1 - semnorm n Lemmas 1 and 2 f p = 0. The next lemma s stated wthout proof due to page restrctons. Lemma 3. Let f W h grad have vanshng nodal values everywhere on Ω except on the wre basket W of Ω. For each subdoman face F of Ω and Ch d H /C, C > 1, there exsts a v W h curl such that v e = f e for all e M F, v e = 0 for all other edges of Ω, and v 2 L 2 (Ω ) C(ω f 2 L 2 ( F) + d2 f t F 2 L 2 ( F)), (10) v 2 L 2 (Ω ) C(τ(d) f 2 L 2 ( F) + f t F 2 L 2 ( F)), (11)

4 18 Clark R. Dohrmann and Olof B. Wdlund where t F s a unt tangent along F, and { 0 f d > H /C τ(d) = d 2 otherwse. The Helmholtz-type decomposton and estmates n the next lemma wll allow us to make use of and buld on exstng tools for scalar functons n H 1 (Ω ). We refer the reader to Lemma 5.2 of [4] for the case of convex polyhedral subdomans; ths mportant paper was preceded by [5], whch concerns other applcatons of the same decomposton. Lemma 4. For a convex and polyhedral subdoman Ω and any u W h curl, there s a q W h curl, Ψ (W h grad )3, and p W h grad such that u = q + Π h (Ψ ) + p, (12) p L 2 (Ω ) C u L 2 (Ω ), (13) Ψ L 2 (Ω ) C u L 2 (Ω ), (14) h 1 q 2 L 2 (Ω ) + Ψ 2 H 1 (Ω ) C u 2 L 2 (Ω ). (15) Lemma 5. For any u W h curl wth u e = 0 for all e M E and E G E, there exsts a v W h curl such that v e = u e for all e M Ω, where v F e = 0 e M Ω \ M F. Further, where the energy E s defned n (3). v = v F, (16) F G F E (v F ) Cω 2 E (u ), (17) Proof. Let p n (12) be chosen so p = 0. Ths s possble snce a constant can be added to p wthout changng ts gradent. Because u e = 0 for all e M E, t follows from Lemmas 1 and 4 and elementary estmates that p t E 2 L 2 (E ) (Π h (Ψ ) + q ) t E 2 L 2 (E ) Cω u 2 L 2 (Ω ). (18) For each subdoman face F of Ω, we fnd from Lemmas 2 and 4 that Defne p F 2 L 2 (Ω ) Cω2 u 2 L 2 (Ω ). (19) { H p W := p V + p E, d := f d H max(d V G V E G,Ch ) otherwse, E where d := α /β. Further, let p W and p F denote the functons f and v, respectvely, of Lemma 3. For each subdoman face F of Ω, we then fnd from Lemmas 1 and 3 and (18) that

5 Some Recent Tools and a BDDC Algorthm for 3D Problems n H(curl) 19 E (p F ) Cω 2 E (u ), (20) where p F e = p W e e M F and p F e = 0 e M Ω \ M F. Wth reference to (12) and (4), we defne q F := q e N e, (21) e M F and from elementary fnte element estmates and Lemma 4 fnd q F 2 L 2 (Ω ) Ch3 e M F q 2 e C q 2 L 2 (Ω ) C u 2 L 2 (Ω ), (22) q F 2 L 2 (Ω ) Ch q 2 e C u 2 L 2 (Ω ). (23) e M F It follows from Lemmas 2 and 4 that there exsts a Ψ F (W h grad )3 such that Ψ F = Ψ at all nodes of F, that vanshes at all other nodes of Ω, and Ψ F 2 L 2 (Ω ) C Ψ 2 L 2 (Ω ) C u 2 L 2 (Ω ), (24) Ψ F 2 H 1 (Ω ) Cω2 Ψ 2 H 1 (Ω ) Cω2 u 2 L 2 (Ω ). (25) From Lemmas 1 and 4, we obtan Ψ 2 L 2 ( F) Cω Ψ 2 H 1 (Ω ) Cω u 2 L 2 (Ω ). (26) Let Ψ F (W h grad )3 be dentcal to Ψ at all nodes of F and vansh at all other nodes of Ω. For g := Π h (Ψ F ), we defne From elementary estmates and (26) we then obtan Defnng g F := g h e N e. (27) e M F g F 2 L 2 (Ω ) Ch2 Ψ 2 L 2 ( F) Cω h 2 u 2 L 2 (Ω ), (28) g F 2 L 2 (Ω ) Cω u 2 L 2 (Ω ). (29) v F := p F + p F + q F + Π h (Ψ F ) + g F, (30) we fnd v F e = u e e M F and v F e = 0 e M Ω \M F. The estmate n (17) then follows from the bounds for each of the terms on the rght-hand-sde of (30) along wth elementary estmates for Π h (Ψ F ). 3 BDDC Background nformaton and related theory for BDDC can be found n several references ncludng [2, 9, 10, 8, 1]. Let u Γ and u Γ denote vectors of fnte element coeffcents assocated wth Γ and Γ. In general, entres n u Γ and u Γj are allowed to dffer for j even though they refer to the same fnte element edge. Entres n the vector ũ Γ are partally contnuous n the sense that specfc edge values or edge averages over certan subsets of Γ are requred to match for adjacent subdomans. In order to obtan consstent entres, we defne the weghted average

6 20 Clark R. Dohrmann and Olof B. Wdlund û Γ = R N j=1r T j D jũ Γj, (31) where R j s a 0-1 (Boolean) matrx that selects the rows of u Γj from u Γ and D j s a dagonal weght matrx wth postve entres. The weght matrces form a partton of unty n the sense that N R T D R = I, (32) =1 where I s the dentty matrx. To summarze, û Γ s fully contnuous whle ũ Γ s only partally contnuous. The number of contnuty constrants that must be satsfed by all the ũ Γ determnes the dmenson of the coarse space. Let S denote the Schur complement assocated wth Γ, whch s defned n (39). The system operator for BDDC s the assembled Schur complement N S = R T S R. (33) =1 From Theorem 25 of [10], the condton number of the BDDC precondtoned operator s bounded above by N κ(m 1 =1 S) sup ût Γ S û Γ ũ Γ N. (34) =1 ũt Γ S ũ Γ Ths remarkably smple expresson shows that the contnuty constrants for ũ Γ should be chosen so that large ncreases n energy do not result from the averagng operaton n (31). For smplcty of notaton, we wll refer to u as the vector of edge fnte element coeffcents for Ω. We have the decomposton u = R T Γ u Γ + R T I u I, (35) where u Γ and u I are vectors of coeffcents assocated wth Γ and the nteror of Ω, respectvely, and each row of R Γ and R I has one nonzero entry of unty. We further decompose u Γ as u Γ = R T F u F + R T E u E (36) F G F E G E = F G F R T F u F + R T W u W (37) = R T W u T W + R T W u W, (38) where W denotes the wre basket for Γ and W = Γ \ W. The Schur complement assocated wth Γ can be expressed as S = A Γ Γ A Γ I A 1 I I A I Γ, (39) where A s the stffness matrx for Ω and A Γ Γ = R Γ A R T Γ, A Γ I = R Γ A R T I, A I I = R I A R T I, etc. (40) Smlarly, for W and F, we ntroduce the Schur complements S W = R W Lemma 5 s now rewrtten n matrx-vector notaton as S R T W, S F = R F S R T F. (41)

7 Some Recent Tools and a BDDC Algorthm for 3D Problems n H(curl) 21 (R F u Γ ) T S F (R F u Γ ) Cω 2 (R W u Γ ) T S W (R W u Γ ). (42) Because of page restrctons, we only consder a very rch coarse space whch ncludes every edge of each subdoman edge. Ths coarse space corresponds to Algorthm C of [11]. In ths case, we have R W u Γ = 0, (43) where u := ũ û, and t follows from (37) and the postve defnteness of S that uγ T S u Γ G F u T F S F u F. (44) F G F Let Ω j denote the subdoman whch shares F wth Ω, and consder the generalzed egenvalue problem S F Φ = S F j ΦΛ, (45) where Φ s a matrx of egenvectors normalzed so that Φ T S F j Φ = I and Λ s a dagonal matrx of postve egenvalues. Introducng the change of varables u F = Φw F, we obtan Choosng we fnd and from (46) and (47) obtan u T F S F u F = w T F Λ m w F, (46) u T jf S F j u jf = w T jf I w jf. (47) ŵ F = ŵ jf = (Λ + I) 1 (Λ w F + w jf ), (48) w F = (Λ + I) 1 ( w F w F ), (49) w jf = (Λ + I) 1 Λ( w jf w F ), (50) u T F S F u F + u T jf S F j u jf 4(ũ T F S F ũ F + ũ T jf S F j ũ jf ). (51) From (44), (51) and (42), we obtan N uγ T S u Γ Cω 2 N ũγ T S ũ Γ, (52) =1 =1 where ω = max1 + log(h /h ). (53) Fnally, from (34), (52), and the trangle nequalty, we obtan Theorem 1 (Condton Number Estmate). The condton number of the BDDC precondtoned operator for ths study s bounded by κ Cω 2. (54)

8 22 Clark R. Dohrmann and Olof B. Wdlund In summary, we have obtaned a favorable condton number estmate that requres no assumptons on the materal propertes of the subdomans. We are unaware of any other algorthms for 3D H(curl) problems wth ths property. Comparng the condton number estmate of Theorem 1 wth those n (1) and (2), we see that the factor of H 2 β /α has been removed and the logarthmc factor has been reduced from four powers to two. We note that the estmate n Theorem 1 also holds for FETI-DP due ts spectral equvalence wth BDDC. The algorthm nvolves a change of varables for edges of each subdoman face, and the choce for ŵ F and ŵ jf n (48) corresponds to the dagonal weght matrces R F D R T F = Λ(Λ + I) 1, (55) R F D j R T F = (Λ + I) 1. (56) We note ths change of varables can be mplemented n practce wth just a few smple modfcatons to the standard BDDC algorthm. Referrng back to the dscusson before (46), the change of varables can be expressed as u Γ = T w Γ. Notce that rows of the square transformaton matrx T for edges not on a subdoman face wll have a sngle dagonal entry of unty snce no change of varables s made for those edges, whle the rows of T correspondng to subdoman face F are obtaned from the matrx of egenvectors Φ appearng n (45). One can then replace D j n (31) by D j := T j D jc Tj 1, where D jc s the dagonal weght matrx assocated wth the new varables (see (56)). In terms of the algorthm n [2], the changes amount to replacng W n (16) and (19) by D and W n (18) and (20) by D T. The mportance of the change of varables for some problems s shown n the next secton. 4 Numercal Results In ths secton, we present some numercal results to verfy the theory and also to provde some addtonal nsghts. The doman s a unt cube dscretzed nto smaller cubc elements. All the examples are solved to a relatve resdual tolerance of 10 8 for random rght-handsdes usng the conjugate gradent algorthm wth BDDC as the precondtoner. The number of teratons and condton number estmates from conjugate gradents are under the headngs of ter and cond n the tables. We consder three dfferent types of weghts for the averagng operator. The frst one, desgnated eg, s the one of the prevous secton based on a change of varables and the soluton of an egenproblem. Unless otherwse specfed n the tables, ths s the weghtng used. The second type, stff, s based on a conventonal approach n whch the weghts are proportonal to entres on the dagonals of subdoman matrces. The thrd, card, uses the nverse of the cardnalty of an edge,.e. the recprocal of the number of subdomans sharng the edge, for the weght. The results n Table 1 are consstent wth theory, suggestng condton numbers are bounded ndependently of the number of subdomans, whle the results n Table 2 are consstent wth the log(h/h) 2 estmate of Theorem 1. We also consder a checkerboard dstrbuton of materal propertes n whch (α, β) for a subdoman s ether (α 1,β 1 ) or (α 2,β 2 ), and note that subdomans wth the same propertes are connected together only at ther corners. Results for 64 cube subdomans each wth H/h = 4 are shown n Table 3. Notce for only one choce of materal propertes n the table

9 Some Recent Tools and a BDDC Algorthm for 3D Problems n H(curl) 23 that all three types of weghtng lead to small condton numbers, and only the eg approach always gves condton numbers whch are ndependent of the materal propertes. We also nvestgated another type of weghtng smlar to card, but wth weghts γ, 0 < γ < 1 for faces of subdomans wth propertes α 1,β 1 and 1 γ for faces of subdomans wth propertes α 2,β 2. Regardless of the choce of γ, large condton numbers were observed for the propertes n the fnal row of Table 3. We note also that the choce of materal propertes n the fnal row s not covered by the theory of [11]. In the fnal example, we consder a cube mesh of 20 3 elements that s parttoned nto dfferent numbers of subdomans usng the graph parttoner Mets [7]. Although ths example s not covered by our theory because the subdomans have rregular shapes, the results n Table 4 ndcate that the algorthm of ths study contnues to perform well. The results n Tables 3 and 4 suggest that the eg weghtng of ths study may be necessary n order to effectvely solve problems wth materal property jumps or wth subdomans havng rregular shapes. Table 1. Results for N cube subdomans, each wth β = 1 and H/h = 4. N α = 10 2 α = 1 α = 10 2 ter (cond) ter (cond) ter (cond) (2.70) 14 (2.63) 10 (1.77) (2.88) 15 (2.81) 11 (2.05) (2.95) 15 (2.87) 12 (2.23) (2.98) 16 (2.91) 13 (2.33) Table 2. Results for 64 cube subdomans, each wth β = 1. H/h α = 10 2 α = 1 α = 10 2 ter (cond) ter (cond) ter (cond) 4 15 (2.70) 14 (2.63) 10 (1.77) 6 17 (3.30) 16 (3.21) 11 (2.14) 8 18 (3.77) 16 (3.66) 13 (2.46) (4.16) 18 (4.03) 13 (2.72) References [1] Susanne C. Brenner and Rdgway Scott. The Mathematcal Theory of Fnte Element Methods. Sprnger-Verlag, Berln, Hedelberg, New York, Thrd edton. [2] Clark R. Dohrmann. A precondtoner for substructurng based on constraned energy mnmzaton. SIAM J. Sc. Comput., 25(1): , 2003.

10 24 Clark R. Dohrmann and Olof B. Wdlund Table 3. Checkerboard materal property results for 64 cube subdomans wth H/h = 4. α 1 β 1 α 2 β 2 eg stff card ter (cond) ter (cond) ter (cond) (1.59) 19 (4.57) 196 (1.64e3) (1.96) 84 (2.69e2) 109 (4.72e2) (2.63) 14 (2.63) 14 (2.63) (1.07) 65 (3.17e2) 74 (1.65e2) Table 4. Results for 20 3 elements parttoned nto N subdomans usng a graph parttoner. Materal propertes are constant wth α = 1 and β = 1. N eg stff card ter (cond) ter (cond) ter (cond) (4.30) 189 (6.31e2) 24 (9.06) (4.40) 184 (6.34e2) 29 (1.55e3) (3.89) 188 (6.47e2) 23 (7.48) (4.16) 176 (6.12e2) 23 (6.49) [3] Clark R. Dohrmann and Olof B. Wdlund. An teratve substructurng algorthm for two-dmensonal problems n H(curl). Techncal Report TR , Department of Computer Scence, Courant Insttute of Mathematcal Scences, New York Unversty, December [4] Ralf Hptmar and Jnchao Xu. Nodal auxlary space precondtonng n H(curl) and H(dv) spaces. SIAM J. Numer. Anal., 45(6): (electronc), [5] Ralf Hptmar, Gsela Wdmer, and Jun Zou. Auxlary space precondtonng n H 0 (curl;ω). Numer. Math., 103(3): , [6] Qya Hu and Jun Zou. A nonoverlappng doman decomposton method for Maxwell s equatons n three dmensons. SIAM J. Numer. Anal., 41(5): , [7] George Karyps and Vpn Kumar. METIS Verson 4.0. Unversty of Mnnesota, Department of Computer Scence, Mnneapols, MN, [8] Jng L and Olof B. Wdlund. FETI DP, BDDC, and Block Cholesky Methods. Internat. J. Numer. Methods Engrg., 66(2): , [9] Jan Mandel and Clark R. Dohrmann. Convergence of a balancng doman decomposton by constrants and energy mnmzaton. Numer. Lnear Algebra Appl., 10(7): , [10] Jan Mandel, Clark R. Dohrmann, and Radek Tezaur. An algebrac theory for prmal and dual substructurng methods by constrants. Appl. Numer. Math., 54: , [11] Andrea Tosell. Dual prmal FETI algorthms for edge fnte element approxmatons n 3D. IMA J. Numer. Anal., 26:96 130, [12] Andrea Tosell and Olof Wdlund. Doman Decomposton Methods - Algorthms and Theory, volume 34 of Sprnger Seres n Computatonal Mathematcs. Sprnger-Verlag, Berln Hedelberg New York, 2005.