A THREE-LEVEL BDDC ALGORITHM FOR SADDLE POINT PROBLEMS

Transcription

1 A TREE-LEVEL BDDC ALGORITM FOR SADDLE POINT PROBLEMS XUEMIN TU Abstract BDDC algoritms ave previously been extended to te saddle point problems arising from mixed formulations of elliptic and incompressible Stokes problems In tese two-level BDDC algoritms, all iterates are required to be in a benign space, a subspace in wic te preconditioned operators are positive definite Tis requirement can lead to large coarse problems, wic ave to be generated and factored by a direct solver at te beginning of te computation and tey can ultimately become a bottleneck An additional level is introduced in tis paper to solve te coarse problem approximately and to remove tis difficulty Tis tree-level BDDC algoritm keeps all iterates in te benign space and te conjugate gradient metods can terefore be used to accelerate te convergence Tis work is an extension of te tree-level BDDC metods for standard finite element discretization of elliptic problems and te same rate of convergence is obtained for te mixed formulation of te same problems Estimate of te condition number for tis tree-level BDDC metods is provided and numerical experiments are discussed Key words BDDC, tree-level, saddle point problem, domain decomposition, coarse problem, benign space, condition number AMS subject classifications 65N3, 65N55, 65F1 1 Introduction Te BDDC Balancing Domain Decomposition by Constraints) metods, wic were introduced and analyzed by Dormann, Mandel, and Tezaur 6, 2, 21, were originally designed for standard finite element discretization of elliptic problems Te BDDC algoritms ave been extended to saddle point problems arising from mixed finite element discretization of Stokes by Li and Widlund 17, to nearly incompressible elasticity by Dormann 7, and to flow in porous media by te autor 27, 29 Te coarse problems, in te BDDC algoritms proposed in 17, 7, 27, 29, are given in terms of a set of primal constraints cosen from eac pair of adjacent subdomains Te matrices of te coarse problems are generated and factored by using a direct solver at te beginning of te computation Te number of selected primal constraints in eac subdomain must be large enoug to make sure tat te iterates stay in te benign space, a special subspace in wic te preconditioned operators are positive definite, see 23, 8, 16, 17, 27 Terefore, te coarse component of te two-level BDDC preconditioner can become a bottleneck if tere are very many subdomains, eg, on a large parallel computing system One way to remove tis difficulty is to use inexact solvers Inexact solvers for iterative substructuring algoritms ave been discussed in 1, 1, 9, 24, 2 Klawonn and Widlund considered inexact solver for te one-level Finite Element Tearing and Interconnect FETI) algoritms in 14 In 32, 31, 3, te autor introduced an additional level into te BDDC algoritms to solve te coarse problem approximately wile maintaining a good convergence rate, see also 28, Capter 3 for details In 19, Li and Widlund considered solving te local problems in te BDDC algoritms approximately by multigrid metods Dormann also as developed several versions of approximate BDDC preconditioners in 5 Klawonn and Department of Matematics, University of California and Lawrence Berkeley National Laboratory, Berkeley, CA xuemin@matberkeleyedu Tis work was supported in part by te Director, Office of Science, Computational and Tecnology Researc, US Deparment of Energy under Contract No DE-AC2-5C

2 Reinbac recently provided and analyzed some inexact Dual-Primal FETI FETI- DP) algoritms in 13 Te tree-level BDDC algoritms ave also been extended to mortar finite element discretization wit geometrically nonconforming partition by Kim and te autor, see 12 Te multi-level BDDC algoritms ave also been studied by Mandel, Sousedik, and Dormann 22 Most of tese inexact solvers are designed for symmetric, positive definite problems In tis paper, we extend te tree-level BDDC metods of 32, 31, 3 to te saddle point problems arising from mixed formulations of elliptic and incompressible Stokes problems For ease of te presentation, we will focus on te formulation and analysis of te mixed formulation of elliptic problems but we will provide numerical results for bot cases It is noteworty tat wang and Cai ave pointed out in 11 tat, te twolevel Scwarz preconditioners wit exact coarse solvers for saddle point problems may not ave optimal computing time In teir numerical experiments, te use of inexact iterative coarse solvers requires less computer time tan te exact one Our tree-level BDDC algoritm provides a way to solve te coarse problem inexactly for two-level BDDC algoritms Tis inexact version of BDDC algoritms requires less memory and is more efficient for parallel computation Moreover, we can also apply several Cebysev iterations to improve te accuracy of tis inexact coarse solver, depending on te application requirement, see 32, 31 It as been establised, by Mandel, Dormann, and Tezaur 21, Li and Widlund 18, and Brenner and Sung 3, tat a pair of te preconditioned FETI-DP and BDDC operators wit te same primal constraints ave te same eigenvalues, except possibly for and 1 owever, an additional coarse level for FETI-DP is not straigtforward since te coarse problem is built into te system matrix Te inexact FETI-DP metods, developed in 13, use block-triangular preconditioners for te saddle point formulation of te FETI-DP metods Teir condition number estimate is based on te existence of a good preconditioner wit optimal spectral bounds for te original FETI-DP coarse matrix We note tat an algebraic multigrid preconditioner is used in 13 and tat te coarse problem will no longer be positive definite wen tis inexact FETI-DP algoritms are applied to saddle point problems, see 16 It appears to be difficult to coose a good preconditioner wit optimal spectral bounds for it ere, we provide an analysis and a good preconditioner for suc coarse problems Tis preconditioner can also be used for te inexact FETI-DP algoritms for saddle point problems Moreover, we can prove tat our tree-level BDDC algoritms will maintain all iterates in te benign space and tat te conjugate gradient metod can be used to accelerate te convergence Te rest of te paper is organized as follows We first review some mixed finite element discretizations and te two-level BDDC metods briefly in Section 2 and Section 3, respectively We introduce our tree-level BDDC metod and te corresponding preconditioner M 1 in Section 4 We give some auxiliary results in Section 5 In Section 6, we provide an estimate of te condition number for te system wit te preconditioner M 1 wic is of te form C 1 + log Ĥ ) log ) 2, were Ĥ,, and are typical diameters of te subregions, subdomains, and elements, respectively; we decompose te wole domain into subregions and eac subregion is ten partitioned into subdomains; see Section 4 for te detail Finally, some computational results are presented in Section 7 2

3 2 An elliptic problem discretized by mixed finite elements We consider te following elliptic problem on a bounded polygonal domain Ω in two dimensions wit a Neumann boundary condition: 21) { a p) f in Ω, n a p) g in Ω ere n is te outward normal to Ω and a is a positive definite matrix function wit entries in L Ω) satisfying ξ T ax)ξ α ξ 2, for ae x Ω and some positive constant α Te functions f L 2 Ω) and g 1/2 Ω) satisfy te compatibility condition Ω fdx + gds Ω Te equation 21) as a solution p wic is unique up to a constant Witout loss of generality, we assume tat g and tat f and te solution p ave mean value zero over Ω We ten ave a unique solution We assume tat we are interested in computing a p directly as is often required in flow in porous media We ten introduce te velocity u a p and call p te pressure We obtain te following system for te velocity u and te pressure p: 22) u a p in Ω, u f in Ω, n u in Ω Let ρx) ax) 1 and define a ilbert space by wit te norm div, Ω) {v L 2 Ω) 2 ; v L 2 Ω)}, v 2 div,ω) v 2 L 2 Ω) + 2 D v 2 L 2 Ω), were D is te diameter of Ω Given a vector u div, Ω), it is possible to define its normal component u n on Ω, as an element of 1/2 Ω) Let div, Ω) be te subspace of div, Ω) defined as: div, Ω) {v L 2 Ω) 2 ; v L 2 Ω) and v n on Ω} Te weak form of 22) is given as follows: find u div, Ω) and p L 2 Ω) {q : q L 2 Ω), Ω qdx } suc tat { au,v) + bv, p) v div, Ω), bu, q) Ω fqdx q L2 Ω), were au,v) Ω ut ρx)vdx and bu, q) u)qdx Ω Let Ŵ be te lowest order Raviart-Tomas finite element space wit a zero normal component on Ω, see 4, Capter I, 3, and let Q be te space of piecewise constants wit a zero mean value Tey are finite dimensional subspaces of div, Ω) and L 2 Ω), respectively Te pair Ŵ, Q satisfy a uniform inf-sup condition, see 4, Capter IV 12 Te finite element discrete problem is: find u Ŵ and p Q suc tat { au,v ) + bv, p ) v Ŵ, bu, q ) Ω fq dx q Q, 3

4 and in matrix form: 23) A B T B u p F Te system matrix of 23) is symmetric indefinite wit te matrix A symmetric, positive definite 3 Te two-level BDDC metod We decompose Ω into N nonoverlapping subdomains Ω i wit diameters i, i 1,, N, and wit max i i We assume tat eac subdomain is a union of sape-regular coarse quadrilaterals and tat te number of suc quadrilaterals forming an individual subdomain is uniformly bounded We also assume tat ρx) is a constant in eac subdomain We ten introduce quasiuniform triangulations of eac subdomain We note tat te results of tis paper are also valid for te finite element spaces based on triangles and te algoritm can be extended to different types of subdomains Te global problem 23) is assembled from te subdomain problems 31) A i) B i)t B i) u i) p i) F i) Let Γ be te interface between te subdomains Te set of te interface nodes Γ is defined as Γ i Ω i, ) \ Ω, were Ω i, is te set of nodes on Ω i and Ω is te set of nodes on Ω We decompose te discrete velocity and pressure spaces Ŵ and Q into Ŵ N i1 W i) I ) ŴΓ and Q N i1 Q i) I ) Q ere, ŴΓ is te space of traces on Γ of functions of Ŵ Te elements of Wi) I are supported in te subdomain Ω i and teir normal components vanis on te subdomain interface Γ i Γ Ω i, wile te elements of Q i) I are restrictions of elements in Q to Ω i wic satisfy Ω i q i) I Q is te subspace of Q wit constant values q i) in te subdomain Ω i tat satisfy N i1 qi) mω i ), were mω i ) is te measure of te subdomain Ω i R i) is te operator wic maps functions in te space Q to its constant component of te subdomain Ω i We denote te subdomain interface velocity variables by W i) Γ and te associate product space by W Γ N i1 Wi) Γ In order to define te BDDC algoritm, we also need to introduce a partially assembled interface velocity space W Γ by ) W N Γ Ŵ W Ŵ i1 W i) ere, Ŵ is te coarse level, primal interface velocity space wic is spanned by subdomain interface edge basis functions wit constant values at te nodes of te edge We cange te variables so tat te degree of freedom of eac primal constraint is explicit, see 18 and 15 Te space W is te direct sum of te W i), wic is spanned by te remaining interface velocity degrees of freedom, wic ave a zero 4

5 average over eac edge In te space W Γ, we ave relaxed most continuity constraints on te velocity across te interface but ave retained all primal continuity constraints, wic as te important advantage tat all te linear systems are nonsingular in te computation We also need to introduce several restriction, injection, and scaling operators between different spaces Te restriction operators are: R i) Γ Ŵ Γ W i) Γ, Ŵ R i) W i), W R i) W i), We also introduce two injection operators: R Γ WΓ W, and W R Γ Γ Ŵ RΓ e R Ŵ Γ Γ WΓ WΓ Te scaled injection operator R D,an be written as R D,Γ D R Γ, were D is a diagonal scaling matrix Te diagonal elements of D, corresponding to te primal variables, are 1, and all oters are given by δ i x) ere, we define te scale factor x) as follows: for γ 1/2, ), δ i 32) δ i x) ρ γ i x) j N x ρ γ j x), x Ω i, Γ, were N x is te set of indices j of te subdomains suc tat x Ω j We ten note tat δ i x) is constant on an edge since te nodes on an edge are sared by te same pair of subdomains We also use te notations RΓ RΓ RD,Γ R, R and RD I I I 33) Te subdomain saddle point problems 31) can be written as A i) B i)t A i)t I B i) B i) A i) I A i) I B i)t B i)t I A i) A i) B i) A i)t I B i) I Ai)T Ai) Bi)T B i)t B i) u i),i p i),i u i), u i), p i), F i),i F i), were u i),i, pi),i,ui),,ui),, pi), ) Wi) I, Qi) I,Wi),Wi), Qi) ) We note tat, by te divergence teorem, te lower left block of te matrix of 33) is zero since te bi-linear form bv i) I, qi) ) always vanises for any vi) I W i) I and any constant q i) in te subdomain Ω i We now reduce te global problem 23) to an interface problem We first introduce te subdomain Scur complements S i) Γ by eliminating te subdomain interior variables u i),i and pi),i in 33): 34) 1 S i) Γ A i) A i) Ai)T I B i)t A i) B i)t A i)t I A i)t I A i) Ai) A i) I B i)t I 5, B i) B i) B i) I

6 Given te definition of S i) Γ, te subdomain problems 33) are reduced to te subdomain interface problems S i) Γ B i)t i) Γ u,γ B i) Γ p i) g i), i 1, 2,, N,, were u i),γ u i), u i),, B i) Γ B i) Bi), and g i) F i), A i) I A i) I B i)t B i)t I A i) B i)t B i) 1 F i),i S Γ Let S 1) Γ S N) Γ, B Γ B 1) Γ B N) Γ, and S SΓ B T Γ B Γ Te partially assembled Scur complement S is obtained from S by assembling te primal variables on te subdomain interface, ie T R 35) S T R SR Γ S ΓR Γ R T SΓ Γ BT BT Γ : Γ B Γ R Γ B Γ S can be furter assembled wit respect to te variables of te W Te fully assembled Scur complement 36) Ŝ R RT T S R Γ SΓ RΓ RT Γ BT Γ ŜΓ BT : Γ B Γ RΓ B Γ and te reduced interface problem can be written as: find u,γ, p, ) ŴΓ Q, suc tat u,γ 37) Ŝ g, p, were g N i1 R i)t Γ R i)t g i) Te two-level preconditioned BDDC algoritm is of te form: find u,γ, p, ) Ŵ Γ Q, suc tat 38) M 1 u,γ Ŝ M 1 g p, 6

7 were te preconditioner M 1 R D T S 1 RD as te following form: 39) R D T R T N T A i) Γ R i) A i) 1 Bi)T A i) I A i) R i) Bi)T R Γ + ΦS 1 i1 B i) B i) ere Φ is te matrix given by 31) R T Γ R T N Γ I R i) i1 T A i) A i) I B i) A i) A i) Bi)T Bi)T B i) Te coarse level problem matrix S is determined by { N R i)t S A i) Bi)T i1 R i)t B i) T A i)t I A i) A i) Bi)T A i)t 311) Bi)T B i) I A i) I B i) A i) Bi)T B i) 1 1 A i)t I A i)t Bi)T B i) I A i)t I A i)t Bi)T B i) I ΦT R i) R i) wic is obtained by assembling subdomain matrices; for additional details, cf 6, 2, 18, 27 We define two subspaces ŴΓ,B and W Γ,B of ŴΓ and W Γ, respectively, as in 17, Definition 1: R D R i) R i), 312) Ŵ Γ,B {w Γ ŴΓ BΓ w Γ }, W Γ,B {w Γ W Γ BΓ w Γ } We call ŴΓ,B Q and W Γ,B Q te benign subspaces of ŴΓ Q and W Γ Q, respectively It is easy to ceck tat bot operators Ŝ and S, given in 36) and 35), are symmetric, positive definite wen restricted to te benign subspaces ŴΓ,B Q and W Γ,B Q, respectively We note tat te solution of 37) is not in te benign subspace owever, we can find a special discrete velocity u,γ ŴΓ and define u Γ u,γ u,γ suc tat te correction u Γ belongs to te benign subspace F 1),Γ u Let p p and,γ p M 1 F N),Γ We obtain u from u,γ by solving subdomain saddle point problem wit Diriclet boundary condition in eac subdomain and let u u u Te correction u, p)t satisfies A B T u Au 313) B p F Bu 7

8 Wit tis special u we coose ere, te divergence of te correction u is not zero, but u Γ is in te benign subspace ŴΓ,B, for details, see 28, Section 48 Te rest of te algoritm is te same as before except tat Au g i) )i) A i) Au I B i)t 1 )i) A i) A i) F i), Bu ) I B i)t B i)t Au I B i) )i) I F i),i Bu )i) I Because of te coice of our primal subspace Ŵ, we ave te following results, see 27, Lemma 41, 17, Lemmas 61 and 62: Lemma 31 B i), for i1,, N For w Γ W Γ, we ave B Γ w Γ B w, were B N i1 Ri)T B i) Ri) and w is te primal part of w Γ Lemma 32 For any w W Γ,B Q, RT D w ŴΓ,B Q Since te correction u Γ, p) T lies in tis benign subspace, we coose te initial guess in te benign subspace and te preconditioned operator defined in 38) will keep all te iterates in tis benign subspace by Lemma 32, in wic te preconditioned operator is positive definite and a preconditioned conjugate gradient metod can be applied We ave te following result for te two-level BDDC algoritm, see 27, Teorem 61: Teorem 1 Te preconditioned operator M 1 Ŝ is symmetric, positive definite wit respect to te bi-linear form, bs on te benign space ŴΓ,B Q and 314) u,u bs M 1 Ŝu,u C 1 + log ) 2 u,u bs bs, u ŴΓ,B Q ere, C is a constant wic is independent of and 4 A tree-level BDDC metod For te tree-level cases, as in 32, 31, 12, te coarse problem matrix S defined in 311) will not be factored exactly Instead, an additional level is introduced and te coarse problem is solved approximately Call te new level te subregion level To distinguis te spaces and operators for te subregion level from tose for te subdomain level, we use te subscript c for te former We decompose Ω into N c subregions Ω j wit diameters Ĥj, j 1,, N c Eac subregion Ω j is te union of N j subdomains Ω j i wit diameters j i Let Ĥ max j Ĥj and max i,j j i, for j 1,, N c, and i 1,, N j Ten N, te total number of subdomains, is given by N N N Nc We assume tat ρx) is a constant in eac subregion We introduce te subregional Scur complement, wic is assembled only from te subdomains in subregion Ω j, Nj S j) i1 41) R i)t R i)t A i)t I A i)t Bi)T B i) I { T A i) B i) A i) A i) I B i) Bi)T A i) A i) Bi)T Bi)T B i) 8 1 A i)t I A i)t Bi)T B i) I R i) R i)

9 By Lemma 31 and 41), we ave B i) and S j) : A j) B j)t B j), were 42) Nj A j) i1 R i)t A i) A i)t I A i)t B i) I T A i) A i) I B i) A i) A i) B i) Bi)T Bi)T 1 A i)t I A i)t B i) I R i), Nj 43) B j) R i)t B i) Ri) i1 We note tat te coarse problem matrix S can be assembled from te S j) Terefore, we can write S as 44) S were A and B are assembled from A j) Recall tat B N i1 Ri)T B i) A B T B Ri), and Bj), respectively, is defined in Lemma 31 By te definition of B and B, we ave Lemma 41 B B In te two-level case, S is factored by a direct solver at te beginning of te computation, cf 39) ere, we build M 1 R T D Replacing S 1 R T N Γ i1 1 in 39) wit M R i) to approximate S 1 gives us te tree-level preconditioner M 1 : T A i) A i) 1 Bi)T A i) I A i) R i) Bi)T R Γ + ΦM 1 B i) B i) To define M 1 in detail, we need to introduce several spaces and operators Let be te interface between te subregions; note tat Γ For eac subregion Ω i, we denote te space corresponding to te subdomain edge average variables in tis subregion, by W c i) Let W c N c i1 Wi) c and let Ŵc be te subspace of W c of elements tat are continuous across W c i) can be decomposed into a subregion interior part W i) and a subregion interface part W i) We furter decompose te subregion interface part W i) into a primal subspace W i) c and a dual subspace W i) c ere, we will only consider te averages over subregion edges as te primal variables Again, we sould cange te variables for all local coarse matrices corresponding to tese edge average primal variables We will assume tat all matrices are written in term of te new variables We denote te associated subregion interface product space by W Γc : N c i1 Wi) We note tat te elements in W Γc can be discontinuous across te subregion interface 9 ΦT R D

10 Ŵ Γc and W Γc are two subspaces of W Γc Te elements of Ŵ are continuous across, wereas only te primal variables are continuous across in W Γc Tus, we ave Ŵ W Γc W Γc We also need two injection operators R Γc and R Γc : RΓc e R Γc Ŵ Γc WΓc WΓc, wic are similar to R Γ and R Γ, respectively Similarly, we also denote by Q i) c, te pressure space of piecewise constant on eac subdomain of te subregion Ω i by Q i) c Let Q c Nc i1 Qi) c Q c can be decomposed into Nc i1 Qi) and Q c, were te elements of Q i) are restrictions of elements in Q c to Ω i wic satisfy q i) Ω i Q c is te subspace of Q c wit constant values q i) c in te subregion Ω i tat satisfy N c i1 qi) c mω i ), were mω i ) is te measure of te subregion Ω i R i) c is te operator wic maps functions in te space Q c to its constant component of te subregion Ω i We also use te notation RΓc RΓc R c and Rc I I We denote by F c and F Γc te dual spaces of Ŵc and Ŵ, respectively We also denote by G c and G c te dual spaces of Q c and Q c, respectively We are now ready to explain ow M 1 works on a vector in F c G c Given Ψ a vector F y c G c, let S 1 Ψ ŷ z and M 1 Ψ ẑ We write Ψ, y, and ŷ in terms of subregion interior and interface parts, ie, Ψ ) T, ) T, ) T Ψ 1),,Ψ Nc),Ψ Γc y y 1),,y Nc),y Γc and ŷ ŷ 1),,ŷ Nc),ŷ Γc ) T, ) T, We also write, z, and ẑ as 1),, Nc), c z z 1),,z Nc),z c ) T and ẑ ẑ 1),,ẑ Nc),ẑ c y y Ψ To obtain, we solve S z by a block factorization Let z R i) : Ŵ W i) be a restriction operator We write A and B given in 44) using subregion interior and interface blocks: and A A 1) IcIc A 1)T ΓcIc R 1) R 1)T A 1) 45) B A Nc) IcIc ΓcIc R Nc)T A Nc) ΓcIc B 1) IcIc B 1) IcΓc R 1) A Nc)T ΓcIc R Nc) Â ΓcΓc B Nc) IcIc B Nc) IcΓc R Nc) BcΓc 1,

11 were  ΓcΓc N c y i) We solve z i) 46) y i) z i) i1 Ri)T A i) ΓcΓc R i) yγc A i) IcIc in terms of B i)t IcIc B i) IcIc and B cγc N c and ave z c 1 Ψ i) i) i1 Ri)T c B i) cγc R i) i) A IcΓc R i) B i) IcΓc R i) yγc z c ) Let T i) A i) ΓcΓc A i) ΓcIc B i)t IcΓc A i) IcIc B i)t IcIc B i) IcIc 1 A i) IcΓc B i) IcΓc and let T i) T i) B i)t cγc B i) cγc, be te subregion Scur complement We ten obtain te subregion interface problem: Nc ) R i)t T i) R i) yγc i1 R i)t c R i) z c c N 1 c ΨΓc R i)t A i) ΓcIc B i)t A i) IcΓc IcIc B i)t IcIc Ψ i) c i1 R i)t c B i) IcIc i) 47) Denote T Γc T 1) T Nc), B c,γc B 1) cγc B Nc) cγc, and T T Γc B T cγc B cγc As on te subdomain level case, we introduce a partially assembled Scur complement of S, and denote it by T T can be written as: T T Γc BT 48) T R c TR c : cγc B cγc T can be furter assembled wit respect to te variables of W i) c T, te fully assembled subregion Scur complement of S, can be written as: 49) T RT c T Rc : T Γc BT cγc B cγc 11

12 Γc We define 41) ΨΓc c N c i1 F Γc G c by c R i)t R i)t c A i) ΓcIc B i)t IcΓc A i) IcIc B i)t IcIc B i) IcIc Te reduced subregion interface problem 47) can be written as: find Ŵ Γc Qc, suc tat 411) T yγc z c ŷ To obtain te approximation ẑ ŷγc Instead, we compute as ẑ c 412) ŷγc ẑ c R c T T R yγc Γc c z c c M 1 Ψ R T D c T 1 RDc Γc c 1 Ψ i) i) yγc z c, we do not solve 411) exactly ere R Dc is a scaled injection operator wic is similar to R D ; we can write R Dc Dc RΓc, were D c is a diagonal scaling matrix Te diagonal elements of D c, I corresponding to te primal variables, are 1, and all oters are given by δ c,i x) ere δ c,i x) is similar to δ i x), wic is defined in 32), except tat δ c,i x) is defined for te subregion interface instead of subdomain interface nodes For an x on te subregion interface, δ c,i x) is defined as follows: for γ 1/2, ), δ c,i x) ρ γ i x) j N x ρ γ j x), were N x is te set of indices j of te subregions suc tat x Ω j Recall tat in our teory, we assume te ρx) are constant in te subregions ŷ i) I ŷγc c y i) ẑ i) and as for ẑ c z i) We will maintain te same relation between yγc and in 46), ie, z c 413) 1 ŷ i) A i) ẑ i) IcIc B i)t IcIc Ψ i) i) B i) IcIc A i) IcΓc R i) B i) IcΓc R i) ) ŷγc ẑ c We also define te subspaces Ŵ,B c and W Γc,B c of Ŵ and W Γc respectively as in 312): Ŵ Γc,B c {w Γc Ŵ BcΓc w Γ }, W Γc,B c {w Γc W Γc BcΓc w Γc } 12

13 We note tat te operators T and T are positive definite in te space Ŵ,B c Q c and W Γc,B c Q c, respectively Now we prove tat our new tree-level preconditioner M 1 can keep all te iterates in te benign space ŴΓ,B Q Lemma 42 For any w ŴΓ,B Q, we ave M 1 Ŝw ŴΓ,B Q Proof: By Lemmas 31, 32 and te definition of M 1, we only need to prove ΦM 1 ΦT RD Ŝw W Γ,B Q Denote w wγ p Since w Γ ŴΓ,B and B i), we ave Φ T f RD Ŝw, were f F ŷ c and recall tat Φ is defined in 31) Let M 1 f ẑ We ŷ obtain by solving 412) and 413) wit Ψ f ẑ and By 41), we know tat c in 412) and tat T 1 Γc RDc W Γc,B c Q c Similarly as te proof of Lemma 32, see 27, Lemma 41 for te details, we ave R D T 1 Γc c T RDc Ŵ,B c Q c, ie, 414) BcΓc ŷ Γc Since, by 413), we ave 415) B i) IcIc ŷ i) + B i) IcΓc R i) ŷ Γc Terefore, by 45), 414), 415), and Lemma 41, we ave B ŷ, ie, B ŷ Finally, we ave ΦM 1 ΦT RD Ŝw ŷ ẑ were denote te dual part of te velocity By Lemma 31, we ave B Γ ŷ B ŷ and we conclude tat ΦM 1 ΦT RD Ŝw W Γ,B Q Remark: In te proof of Lemma 42, B i) is te key ingredient, wic is obtained by coosing enoug primal constraints in eac subdomain Similarly, for te Stokes problem, suc a result is given in 17, Lemma 61; it is based on 17, Assumption 1 Terefore, Lemma 42 is valid for Stokes problem wit 17, Assumption 1 eld Our numerical experiments for Stokes in Section 7 are also consistent wit tis As for te two-level BDDC metod, we coose te initial guess in te benign subspace and te tree-level BDDC preconditioned operator M 1 Ŝ will by Lemma 42 keep all te iterates in tis benign subspace, in wic te preconditioned operator is positive definite and a preconditioned conjugate gradient metod can be applied 13

14 5 Some auxiliary results In tis section, we will collect a number of results wic are needed in our teory In order to avoid a proliferation of constants, we will use te notation A B Tis means tat tere are two constants c and C, independent of any parameters, suc tat ca B CA, were C < and c > In our teory, we make an assumption for our decomposition of te global domain Ω Assumption 51 Eac subdomain Ω i is quadrilateral Te subdomains form a quasi-uniform coarse mes of Ω wit mes size We list some results for Raviart-Tomas finite element function spaces needed in our analysis Tese results were originally given in 26, 34, 25, 33 We define an te interpolation operator RT from Ŵ onto Ŵ, were Ŵ is te Raviart-Tomas finite element space on te coarse mes wit mes size Ŵ is defined in terms of te degrees of freedom λ E over te edges E in te coarse mes, by λ E RTu) : 1 u nds E E We consider te stability of te interpolant RT in te next lemma Lemma 52 Tere exists a constant C, wic depends only on te aspect ratios of K T and of te elements of T, suc tat, for all u Ŵ, div RT u) 2 L 2 K) divu 2 L 2 K), RT u 2 L 2 K) C 1 + log ) ) u 2 L 2 K) + 2 K divu 2 L 2 K) Proof: See 26, Lemma 32 We next introduce tree useful extension operators We define N Ω i ) as te te space of functions tat are constant on eac element of te edges of te boundary of Ω i and its subspace N Ω i ), of functions tat ave mean value zero on Ω i Definition 1 Given te boundary data wit zero mean value on Ω i, define a discrete divergence free extension operator by solving te local saddle point problem 33) wit u i), u i), given and zero rigt and side Te resulting u i) as te minimal L 2 -norm of all discrete divergence free velocities wit te given boundary values, ie, i : N Ω i ) W i) satisfies B i) i µ and i µ L2 Ω i) min v L2 Ω i), v W i), B i) v, v Ωi µ for any µ N Ω i ) Definition 2 Given edge average values over te subdomain edges wit a zero mean value over Ω i, define a discrete divergence free extension operator Ĥi by solving te local saddle point problem 33) wit given u i), and zero rigt and side Te resulting velocity u i) as minimal L2 -norm of all discrete divergence free velocities wit given edge averages, ie, Ĥ i : N Ω i ) W i) satisfies B i) Ĥ i µ and Ĥiµ L2 Ω i) min v L2 Ω i), v W i), B i) v, v Eµ 14

15 for any µ N Ω i ), Definition 3 Given te values w i) i by obtaining w i) i i) : W,B c W c i) using 46) wit R i) y Γc satisfies: W i),b c, define te extension operator w i) y i), ie, and wit w i) B i) i w Γc, and i w Γc A i) min v W i) c, B v, v Ω iw i) Γc v i) A µ 2 S i) Γ From te definitions of i, Ĥ i, and i, we ave te following lemma: Lemma 53 i µ 2 L 2 Ω i), µ 2 A i) N i j1 ĤjR i) µ) 2 L 2 Ω i j ), wi) 2 T i) Γc i w Γc 2 A i) In next two lemmas, we establis te relation of te L 2 -norm of te functions in Ŵ and teir Ĥ extensions Lemma 54 Let D be a square wit vertices A, ), B, ), C, ), and D, ), wit a quasi-uniform triangulation of mes size Ten, tere exists a divergence free Raviart Tomas finite element function v defined on D suc tat v 2 L 2 D) C1 + log ) and RT v n ) AB log, RT v n AD ) log, RT v n BC RT v n CD Proof: Using 32, Lemma 41, we can construct a discrete armonic piecewise bilinear finite element function ϕ in D wit te following properties: ϕ 2 1 D) 1 + log, ϕ L D) ϕa) 1 + log, ϕb) ϕc) ϕd) Let v ϕ y, ϕ x ) We ave divv and v 2 L 2 D) ϕ 2 1 D) 1 + log, and RT v n AB 1 RTv n DA 1 ϕ x x, ) dx log ), ϕ y, y) dy log ) Remark: In Lemma 54, we ave constructed te function v for te square D By using similar ideas, we can easily construct a function v, wic satisfies te same properties, for oter sape-regular quadrilaterals wic can be obtained from te reference square by a sufficiently benign mapping Lemma 55 Under Assumption 51, let Wi be te lowest order Raviart-Tomas finite element function space on a subregion Ω i wit a quasi-uniform coarse mes wit mes size And let Wi,j, j 1,, N i, be te lowest order Raviart-Tomas finite element function space on a subdomain Ω i j wit a quasi-uniform fine mes wit mes 15

16 size Given a discrete divergence free u W i, ie, u n wit a zero mean value on Ω i j, tere exist two positive constants C 1 and C 2, wic are independent of Ĥ,, and, suc tat C 1 1+log N i ) Ĥju) 2 L 2 Ω i j ) u 2 L 2 Ω i ) C 21+log N i ) Ĥju) 2 L 2 Ω, i j ) j1 j1 were Ĥj is defined in Definition 2 for eac subdomain Ω i j, j 1,, N i Proof: Witout loss of generality, we assume tat te subdomains are squares Denote te edges of te subdomain Ω i j by a j, b j, c j, and d j, and denote te normal components of u at tese four edges by u n aj ua j ), u n bj ub j ), u n cj uc j ), and u n dj ud j ), respectively We ave ua j ) ub j )+uc j )+ud j ) since u as zero mean value over Ω i j u is a lowest order Raviart-Tomas finite element function and we ave, u 2 L 2 Ω i ) N i j1 u 2 L 2 Ω i j ) and 51) u 2 L 2 Ω i j ) C2 u 2 a j ) + u 2 b j ) + u 2 c j ) + u 2 d j )) ) According to Lemma 54, we can construct divergence free functions φ b, φ c, and φ d on Ω i j suc tat RTφ b n bj ub j ) log ), RTφ b n cj RTφ b n bj, RTφ b n dj RTφ b n aj ; RTφ c n cj ub j ) + uc j )) log ), RTφ c n dj RTφ c n cj, RTφ c n aj RTφ c b j ) n bj ; RTφ d n dj ub j ) + uc j ) + ud j )) log ), RTφ d n aj RTφ d n dj, RTφ d n bj RTφ d n cj ; and wit φ b 2 L 2 Ω i j ) ub j) log ), φ c 2 L 2 Ω i j ) ub j) + uc j )) log ), φ d 2 L 2 Ω i j ) ub j) + uc j ) + ud j )) log ) 52) 16

17 Let v j φ 1+log b +φ c +φ d ); we ten ave RT v jm j ) um j ), m a, b, c, d, and ) 2 v j 2 L 2 Ω i j ) 1 + log φ b + φ c + φ d 2 L 2 Ω i j ) ) log φ m 2 L 2 Ω i j ) mb,c,d ) 2 1 C c 1/2 1 + log ) 1 + log ) 2 um j )) 2 53) 1 C log ) u 2 L 2 Ω i j ) ere, we ave used 51) and 52) for te last two inequalities By te definition of Ĥ j u), we ave, ma,b,c,d Ĥju) 2 L 2 Ω i j ) v j 2 L 2 Ω i j ) 1 C log ) u 2 L 2 Ω i j ) Summing over all te subdomains of te subregion Ω i, we ave, C log ) N i N i Ĥju) 2 L 2 Ω i j ) u 2 L 2 Ω i j ) u 2 L 2 Ω i ) j1 Tis proves te first inequality We prove te second inequality by noting tat u RT Ĥu)) Since div Ĥu), we ave te second inequality by Lemma 52 We next list several results for te two-level BDDC metods To be fully rigorous, we assume tat eac subregion is a union of sape-regular coarse quadrilaterals and tat te number of suc quadrilaterals forming an individual subregion is uniformly bounded Moreover te fine triangulation of eac subdomain is quasi uniform Under Assumption 51, we can ten get uniform constants C 1 and C 2 in Lemma 55, wic old for all te subregions We define subregion interface averaging operator E Dc R c RT Dc and E Dc, R Γc RT Dc,, wic computes te velocity averages across te subregion interface and ten distributes te averages to te boundary points of te subregions We note tat te estimate of E Dc is te key to te analysis of our tree-level BDDC algoritms We ave for any vector w c w Γc, q c ) W Γc Q c, wγc EDc,Γ 54) E Dc c w Γc, q c q c see 27, 53) for more details Te interface averaging operator E Dc satisfies te following bound, see 27, Lemma 56: Lemma 56 For te two-level BDDC, we ave E Dc u c 2 e S C 1 + log Ĥ j1 ) 2 u c 2 es, u c W Γ,B Q, 17

18 were S and W Γ,B Q, wic corresponds to a mes size, are analogous to S and W Γ,B Q, wic corresponds to a mes size, respectively We next list some results for te subspace W Γ,B Q and W Γc,B c Q c Let w W Γ,B Q, w 2 e S w T Sw, and wγ 2 e SΓ w T Γ S Γ w Γ Similarly, let w c W Γc,B c Q c, w c 2 e T w T Tw, and wγc 2 e TΓc w T TΓc w Γc As in 27, Lemma 55, we ave te following result: Lemma 57 Given any w W Γ,B Q and w c W Γc,B c Q c, we ave w 2 e S w Γ 2 e SΓ and w c 2 e T w Γc 2 e TΓc In addition, we ave, wit T defined in 48): Lemma 58 wγc E Dc w c 2 e T C 1 + log Ĥ ) 2 w c 2 et, for any w c q W Γc,B c Q c ere C is a constant independent of Ĥ,, c, and ρ Proof: Te proof is similar to tat of 32, Lemma 43 Denote by i and i on te mes size ), te extensions in eac subregion Ω i, and by Ĥi j te extension in eac subdomain Ωi j, were i 1,, N, and j 1,, N i We recall tat,, and Ĥ are defined in Definitions 3, 1, and 2, respectively Let R i) be a restriction operator from W Γc to W i) Using te definitions of, Ĥ,, and Lemma 53, we ave E Dc w c 2 T e E Dc, w Γc 2 TΓc e i R i) E Dc, w Γc ) 2 A i) i1 N i ) ρ i Ĥi i j R i) E Dc, w Γc ) i1 j1 By Lemmas 55, 53, and 57, E Dc w c 2 T e i1 ρ i N i Ĥi j j1 1 C log ) 2 L 2 Ω i j ) ) i R i) E Dc, w Γc ) i1 1 C log ) E D c w c 2 e S i1 2 L 2 Ω i j ) ) ) ρ i i R i) E Dc, w Γc ) 2 L 2 Ω i ) 18 i R i) E Dc, w Γc ) 2 A i)

19 Using Lemmas 56, 57, and 53, we obtain E Dc w c 2 1 T e C log ) E D c w c 2 S e ) 2 C C log ) 1 + log Ĥ w c 2 es ) 2 C C log ) 1 + log Ĥ Nc i1 ) 2 C C log ) 1 + log Ĥ Nc i1 ) ) ρ i i R i) w Γc ) 2 L 2 Ω i ) ) ) ρ i i R i) w Γc ) 2 L 2 Ω i ) By Lemmas 55, 53, and 57, we ave E Dc w c 2 e T C C log ) 1 + log Ĥ C C log ) i1 CC 2 C 1 CC 2 C 1 ρ i N i j1 1 + log Ĥ ) 2 Nc Ĥi j i R i) w Γc ) ) log Ĥ Nc i1 1 + log Ĥ i1 ) ) ρ i i R i) w Γc ) 2 L 2 Ω i ) ) 2 C log ) ) 2L 2Ω ij) ) 2 w Γc 2 et Γc CC 2 C 1 ) ) i R i) w Γc ) 2 A i) 1 + log Ĥ ) 2 w c 2 et Ψ Lemma 59 Given any u ŴΓ,B Q, let Φ T RD Ŝu We ave, Ψ T S 1 Ψ Ψ T M 1 Ψ C 1 + log Ĥ ) 2 Ψ T S 1 Ψ Proof: Since u ŴΓ,B Q, we ave as in Lemma 42 19

20 Using 46), 41), and 411), we ave Ψ i) Ψ i1 T S 1 Ψ + Γc + i1 i1 Ψ i)t Ψ i)t T A i) IcIc i1 i1 Ψ i)t y i) + Ψ T y Γc B i)t IcIc B i) IcIc R i)t A i) IcIc A i) ΓcIc B i)t IcIc B i) IcIc A i) IcIc B i)t IcIc B i) IcIc 1 Ψ i) B i)t IcΓc 1 i) Ψ 1 i) Ψ A i) IcIc i) A IcΓc B i) IcΓc B i)t IcIc B i) IcIc + T y Γc + Γc R i) y Γc 1 Ψ i) ) T y Γc T RT c T R c ) 1 Γc Using 413), 41), and 412), we also ave Ψ i) Ψ i1 T M 1 Ψ + Γc + i1 i1 Ψ i)t Ψ i)t T A i) IcIc i1 i1 Ψ i)t ŷ i) + Ψ T ŷ Γc B i)t IcIc B i) IcIc R i)t A i) IcIc A i) ΓcIc B i)t IcIc B i) IcIc A i) IcIc B i)t IcIc B i) IcIc 1 Ψ i) B i)t IcΓc 1 i) Ψ 1 i) Ψ A i) IcIc i) A IcΓc B i) IcΓc B i)t IcIc B i) IcIc + T ŷ Γc + Γc T Γc We only need to compare RT T R ) 1 Γc c c for any Γc F Γc Γc Let and let R i) ŷ Γc ) 1 Ψ i) T ŷ Γc T RT D c T 1 RDc Γc and Γc T RT D c T 1 RDc Γc 55) w c RT c T Rc ) 1 Ŵ Γc,B c Q c and v c T 1 RDc W Γc,B c Q c Following te proofs for 32, Lemma 46 and 31, Lemma 47, we can prove: T RT Dc T 1 RDc ) C ) log Ĥ T RT c T R ) ) 1 c 2

21 Table 1 Condition number bounds and iteration counts wit te two-level preconditioner M 1 and te tree-level preconditioner M f 1 N is te number of te subdomains used for M 1 and N c is te number of te subregions used for M f 1 Ĥ 4 and 4 We keep te same numbers of te subdomains and te same subdomain local problem size for tese two preconditioners ρ 1 Exact Inexact N Cond Iter N c Cond Iter Out of Memory Table 2 Condition number bounds and iteration counts wit te preconditioner f M 1 wit a cange of te number of subdomains and te size of subdomain problems wit N c 4 4 subregions ρ subregions, fixed 4 4 subregions, N fixed b Cond Iter Cond Iter Condition number estimate for te new preconditioner In order to estimate te condition number for te system wit te new preconditioner M 1, we compare it to te system wit te preconditioner M 1 Lemma 61 Given any u ŴΓ,B Q, 61) u T M 1 Ŝu u T M 1 Ŝu C 1 + log Ĥ ) 2 u T M 1 Ŝu Proof: We obtain our result by calculating u T M 1 Ŝu and u T M 1 u for any u Ŵ Γ,B Q and using Lemma 59 Teorem 62 Te condition number for te system wit te tree-level preconditioner M 1 is bounded by C1 + log Ĥ )2 1 + log )2 Proof: Combining te condition number bound, given in 314), for te two-level BDDC metod, and Lemma 61, we find tat te condition number for te tree-level metod is bounded by C1 + log Ĥ )2 1 + log )2 7 Numerical experiments We ave applied our tree-level BDDC algoritm to te model problem 21), were Ω, 1 2 We decompose te unit square into N N subregions wit te side-lengt Ĥ 1/ N and eac subregion into N N subdomains wit te side-lengt Ĥ/N Equation 21) is discretized, in eac subdomain, by te lowest order Raviart-Tomas finite elements and te space of piecewise constants wit a finite element diameter, for te velocity and pressure, 21

22 Table 3 Condition number bounds and iteration counts wit te two-level preconditioner M 1 and te tree-level preconditioner M f 1 N is te number of te subdomains used for M 1 and N c is te number of te subregions used for M f 1 Ĥ 4 and 4 We keep te same numbers of te subdomains and te same subdomain local problem size for tese two preconditioners ρ is in a ceckerboard pattern Exact Inexact N Cond Iter N c Cond Iter Out of Memory Table 4 Condition number bounds and iteration counts wit te preconditioner f M 1 wit a cange of te number of subdomains and te size of subdomain problems wit N c 4 4 subregions ρ is in a ceckerboard pattern 4 4 subregions, fixed 4 4 subregions, N fixed b Cond Iter Cond Iter Table 5 Stokes) Condition number bounds and iteration counts wit te two-level preconditioner M 1 and te tree-level preconditioner M f 1 N is te number of te subdomains used for M 1 and N c is te number of te subregions used for M f 1 Ĥ 4 and 4 We keep te same numbers of te subdomains and te same subdomain local problem size for tese two preconditioners Exact Inexact N Cond Iter N c Cond Iter Out of Memory Table 6 Stokes) Condition number bounds and iteration counts wit te preconditioner fm 1 wit a cange of te number of subdomains and te size of subdomain problems wit N c 4 4 subregions 4 4 subregions, fixed 4 4 subregions, N fixed b Cond Iter Cond Iter

23 respectively Te preconditioned conjugate gradient iteration is stopped wen te l 2 -norm of te residual as been reduced by a factor 1 6 We ave carried out two different sets of experiments to obtain iteration counts and condition number estimates All te experimental results are fully consistent wit our teory In te first set of experiments, we take te coefficient ρ 1 Table 1 gives te iteration counts and condition number estimates for te two-level and te tree-level BDDC algoritms We use te same numbers of subdomains for bot two and tree levels wit a cange of te number of subregions for te tree level case We find tat te condition numbers are independent of te number of subregions And wen te number of te subdomains is large, te two-level algoritms is out of memory, wile te tree-level algoritm is still fine Table 2 gives results wit a cange of te number of te subdomains in eac subregion and a cange of te size of te subdomain problem In te second set of experiments, we take te coefficient ρ 1 in alf of te subregions and ρ 1 in te neigboring subregions, in a ceckerboard pattern Everyting else is te same as in te first set of experiments Te results are reported in Table 3 and 4 Finally, we ave applied our tree-level BDDC algoritm to incompressible Stokes equations We use te same discretization as in 17, were te two-level BDDC algoritm was applied for Stokes Table 5 gives te iteration counts and condition number estimates for te two-level and te tree-level BDDC algoritms We use te same numbers of te subdomains for bot two and tree levels wit a cange of te number of subregions for te tree level case Te similar results are obtained for Stokes as for te elliptic problems Table 6 gives results wit a cange of te number of te subdomains in eac subregion and a cange of te size of te subdomain problem Tese numerical results are also consistent wit our condition number estimates Acknowledgments Te autor is grateful to Professor Olof Widlund for all te elp and time e as devoted to my work REFERENCES 1 Cristop Börgers Te Neumann Diriclet domain decomposition metod wit inexact solvers on te subdomains Numer Mat, 55: , James Bramble, Josep E Pasciak, and Apostol Vassilev Non-overlapping domain decomposition preconditioners wit inexact solves In Domain Decomposition Metods in Sciences and Enginering IX, pages 4 52, Susanne C Brenner and Li-Yeng Sung BDDC and FETI-DP witout matrices or vectors Comput Metods Appl Mec Engrg, 1968): , 27 4 Franco Brezzi and Micel Fortin Mixed and ybrid finite element, volume 15 of Springer Series in Computational Matematics Springer Verlag, Berlin-eidelberg-New York, C R Dormann An approximate BDDC preconditioner Numer Linear Algebra Appl, 142): , 27 6 Clark R Dormann A preconditioner for substructuring based on constrained energy minimization SIAM J Sci Comput, 251): , 23 7 Clark R Dormann A substructuring preconditioner for nearly incompressible elasticity problems Tecnical Report SAND , Sandia National Laboratories, Albuquerque, New Mexico, October 24 8 Paulo Goldfeld, Luca F Pavarino, and Olof B Widlund Balancing Neumann-Neumann preconditioners for mixed approximations of eterogeneous problems in linear elasticity Numer Mat, 952): , 23 9 Gundolf aase, Ulric Langer, and Arnd Meyer Te approximate Diriclet domain decomposition metod I An algebraic approac Computing, 472): , Gundolf aase, Ulric Langer, and Arnd Meyer Te approximate Diriclet domain decom- 23

24 position metod Applications to 2nd-order elliptic BVPs Computing, 472): , Feng-Nan wang and Xiao-Cuan Cai Parallel fully coupled Scwarz preconditioners for saddle point problems Electron Trans Numer Anal, 22: electronic), yea yun Kim and Xuemin Tu A tree-level BDDC algoritm for mortar discretization Tecnical Report LBNL-62791, Lawrence Berkeley National Laboratory, June Axel Klawonn and Oliver Reinbac Inexact FETI-DP metods Internat J Numer Metods Engrg, 692):284 37, Axel Klawonn and Olof B Widlund A domain decomposition metod wit Lagrange multipliers and inexact solvers for linear elasticity SIAM J Sci Comput, 224): , October 2 15 Axel Klawonn and Olof B Widlund Dual-primal FETI metods for linear elasticity Comm Pure Appl Mat, 5911): , Jing Li A dual-primal FETI metod for incompressible Stokes equations Numer Mat, 12: , Jing Li and Olof B Widlund BDDC algoritms for incompressible Stokes equations SIAM J Numer Anal, 446): , Jing Li and Olof B Widlund FETI DP, BDDC, and block Colesky metods Internat J Numer Metods Engrg, 66:25 271, Jing Li and Olof B Widlund On te use of inexact subdomain solvers for BDDC algoritms Comput Metods Appl Mec Engrg, 1968): , 27 2 Jan Mandel and Clark R Dormann Convergence of a balancing domain decomposition by constraints and energy minimization Numer Linear Algebra Appl, 17): , Jan Mandel, Clark R Dormann, and Radek Tezaur An algebraic teory for primal and dual substructuring metods by constraints Appl Numer Mat, 542): , Jan Mandel, Bedric Sousedik, and Clark R Dormann On multilevel BDDC Tecnical Report CCM Report 237, Center for Computational Matematics, University of Colorado at Denver, October 26 To appear in Computing 23 Luca F Pavarino and Olof B Widlund Balancing Neumann-Neumann metods for incompressible Stokes equations Comm Pure Appl Mat, 553):32 335, Marc Barry F Smit A parallel implementation of an iterative substructuring algoritm for problems in tree dimensions SIAM J Sci Comput, 142):46 423, Marc Andrea Toselli Domain decomposition metods for vector field problems PD tesis, Courant Institute of Matematical Sciences, May 1999 TR-785, Department of Computer Science 26 Andrea Toselli, Olof B Widlund, and Barbara I Wolmut An iterative substructuring metod for Maxwell s equations in two dimensions Mat Comp, 7235): , Xuemin Tu A BDDC algoritm for a mixed formulation of flows in porous media Electron Trans Numer Anal, 2: , Xuemin Tu BDDC Domain Decomposition Algoritms: Metods wit Tree Levels and for Flow in Porous Media PD tesis, Courant Institute, New York University, January 26 TR25-879, Department of Computer Science, Courant Institute ttp://csnyuedu/csweb/researc/tecreports/tr25-879/tr25-879pdf 29 Xuemin Tu A BDDC algoritm for flow in porous media wit a ybrid finite element discretization Electron Trans Numer Anal, 26:146 16, 27 3 Xuemin Tu Tree-level BDDC In Domain decomposition metods in science and engineering XVI, volume 55 of Lect Notes Comput Sci Eng, pages Springer, Berlin, Xuemin Tu Tree-level BDDC in tree dimensions SIAM J Sci Comput, 294): , Xuemin Tu Tree-level BDDC in two dimensions Internat J Numer Metods Engrg, 69:33 59, Barbara I Wolmut Discretization Metods and Iterative Solvers Based on Domain Decomposition, volume 17 of Lecture Notes in Computational Science and Engineering Springer Verlag, Barbara I Wolmut, Andrea Toselli, and Olof B Widlund Iterative substructuring metod for Raviart-Tomas vector fields in tree dimensions SIAM J Numer Anal, 375): , 2 24