TR THREE-LEVEL BDDC IN THREE DIMENSIONS

Transcription

1 R REE-LEVEL BDDC IN REE DIMENSIONS XUEMIN U Abstract BDDC methods are nonoverlapping iterative substructuring domain decomposition methods for the solution of large sparse linear algebraic systems arising from discretization of elliptic boundary value problems Its coarse problem is given by a small number of continuity constraints which are enforced across the interface he coarse problem matrix is generated and factored by direct solvers at the beginning of the computation and it can ultimately become a bottleneck, if the number of subdomains is very large In this paper, two three-level BDDC methods are introduced for solving the coarse problem approximately in three dimensions his is an extension of previous work for the two dimensional case and since vertex constraints alone do not suffice to obtain polylogarithmic condition number bound, edge constraints are considered in this paper Some new technical tools are then needed in the analysis and this makes the three dimensional case more complicated than the two dimensional case Estimates of the condition numbers are provided for two three-level BDDC methods and numerical experiments are also discussed Key words BDDC, three-level, three dimensions, domain decomposition, coarse problem, condition number, Chebyshev iteration AMS subject classifications 65N30, 65N55 1 Introduction BDDC Balancing Domain Decomposition by Constraints methods, which were introduced and analyzed in [3, 7, 8], are similar to the balancing Neumann-Neumann algorithms owever, the coarse problem, in a BDDC algorithm, is given in terms of a set of primal constraints and the matrix of the coarse problem is generated and factored by using direct solvers at the beginning of the computation We note that there are now computer systems with more than 100,000 powerful processors, which allow very large and detailed simulations he coarse component of a two-level preconditioner can therefore be a bottleneck if the number of subdomains is very large One way to remove this difficulty is to introduce one or more additional levels In our recent paper [11], two three-level BDDC methods were introduced for two dimensional problems with vertex constraints We solve the coarse problem approximately, by using the BDDC idea recursively, while a good rate of convergence still can be maintained owever, in three dimensional space, vertex constraints alone are not enough to obtain good polylogarithmic condition number bound due to much weaker interpolation estimate and constraints on the averages over edges or faces are needed he new constraints lead to a considerably more complicated coarse problem and the need for new technical tools in the analysis In this paper, we extend the two three-level BDDC methods in [11] to the three dimensional case using primal edge average constraints With the help of the new technical tools, we provide estimates of the condition number bounds of the system with these two new preconditioners he rest of the paper is organized as follows We first review the two-level BDDC methods briefly in Section 2 We introduce our first three-level BDDC method and the corresponding preconditioner M 1 in Section 3 We give some auxiliary results in Section 4 In Section 5, we provide an estimate of the condition number bound for the system with the preconditioner M 1 which is of the form C 1 + log Ĥ log h where Ĥ,, and h are typical diameters of the subregions, subdomains, and elements, 2, Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012, USA xuemin@cimsnyuedu his work was supported in part by the US Department of Energy under Contracts DE-FG02-92ER25127 and DE-FC02-01ER

2 respectively In Section 6, we introduce a second three-level BDDC method which uses Chebyshev iterations We denote the corresponding preconditioner by M 1 We show that the condition number bound of the system with the preconditioner M 1 is of the form CCk 2, 1 + log h where Ck is a function of k, the number of Chebyshev iterations, and also depends on the eigenvalues of the preconditioned coarse problem, and the two parameters chosen for the Chebyshev iteration Ck goes to 1 as k goes to, ie, the condition number approaches that of the two-level case Finally, some computational results are presented in Section 7 2 he two-level BDDC method We will consider a second order scalar elliptic problem in a three dimensional region Ω: find u 0 1 Ω, such that 21 ρ u v = fv Ω Ω v 1 0Ω, where ρx > 0 for all x Ω We decompose Ω into N nonoverlapping subdomains Ω i with diameters i, i = 1,, N, and set = max i i We then introduce a triangulation for all the subdomains Let e the interface between the subdomains and the set of interface nodes Γ h is defined by Γ h = i j Ω i,h Ω j,h \ Ω h, where Ω i,h is the set of nodes on Ω i and Ω h is the set of nodes on Ω he nodes of the different triangulations match across Γ Let W i be the standard finite element space of continuous, piecewise trilinear functions on Ω i ; the algorithms and theories developed in this paper work for other lower order finite elements as well We assume that these functions vanish on Ω Each W i can be decomposed into a subdomain interior part W i I and a subdomain interface part W i Γ, ie, Wi = W i i I W Γ, where the subdomain interface part W i Γ will be further decomposed into a primal subspace Wi and a dual subspace i, ie, Wi W W i Γ = Wi We denote the associated product spaces by W := N Wi, W Γ := N Wi Γ, W := N Wi, W := N Wi, and W I := N Wi I Correspondingly, we have W = W I WΓ, and W Γ = W W We will consider elements of W which are discontinuous across the interface owever, the finite element approximation of the elliptic problem is continuous across Γ We denote the corresponding subspace of W by Ŵ We further introduce an interface subspace W Γ W Γ, for which certain primal constraints are enforced ere, we only consider the case of edge average constraints over all the edges of all subdomains We change the variables to make the edge average degrees of freedom explicit, see [4, Sec 62] and [5, Sec 23] From now on, we assume all the matrices are written in terms of the new variables he continuous primal subspace denoted by Ŵ is spanned by the continuous edge average variables of each edge of the interface he space W Γ can be decomposed into W Γ = Ŵ W he global problem has the form: find u I,u,u W I,Ŵ,Ŵ, such that A II A A I A A A A I A A 2 u I u u = f I f f

3 his problem is assembled from the subdomain problems A i II A i A i I A i A i A i A i I Ai Ai u i I u i u i = We also denote by F Γ, F Γ, and F Γ, the right hand side spaces corresponding to W Γ, Ŵ Γ, and W Γ, respectively In order to describe BDDC algorithm, we need to introduce several restriction, extension, and scaling operators between different spaces he restriction operator R i Γ maps a vector of the space ŴΓ to a vector of the subdomain subspace W i Γ Each column of R i Γ with a nonzero entry corresponds to an interface node, x Ω i,h Γ h, shared by the subdomain Ω i and its next neighbor subdomains R i Γ is similar to R i and represents the restriction from W Γ to W i Γ Ri matrix which extracts the subdomain part, in the space W i f i I f i f i Γ, : W W i, is the restriction, of the functions in the Multiplying space W R i is the restriction operator from the space Ŵ to W i each such element of R i Γ, Ri Γ, and Ri with δ i x gives us Ri D,Γ, Ri D,Γ respectively ere, we define δ i x as follows: for γ [1/2,, δ i x =, and Ri D,, ρ γ i x Pj Nx x, ργ j x Ω i,h Γ h, where N x is the set of indices j of the subdomains such that x Ω j and ρ j x is the coefficient of 21 at x in the subdomain Ω j Furthermore, R Γ and R Γ are the restriction operators from the space W Γ onto its subspace W and W respectively R Γ : ŴΓ W Γ and R Γ : W Γ W Γ are the direct sums of R i Γ R i Γ, respectively RΓ : ŴΓ W Γ is the direct sum of R Γ and the R i scaled operators R D,Γ and R D, are the direct sums of R i D,Γ and R Γ he and Ri D,, respectively R D,Γ is the direct sum of R Γ and R D, R Γ We also use the same restriction, extension, and scaled restriction operators for the right hand side spaces F Γ, F Γ, and F Γ We define an operator S Γ : W Γ F Γ, which is of the form: given u Γ = u u Ŵ W = W Γ, find S Γ u Γ F y eliminating the interior variables of the system: 22 = A u 1 I,u 1,u N,u,,uN I 0, R 1 R Γ S Γ u Γ,,0, R N R Γ S Γ u Γ, R Γ SΓ u Γ, where A is of the form A 1 II A 1 A 1 A 1 A 1 I R1 A 1 R1 A N II A N R 1 A1 I R 1 A1 RN A N I 3 A N A N R N A N A N I R N A i Ri N Ri Ai Ri

4 he reduced interface problem can be written as: find u Γ ŴΓ such that R Γ S Γ RΓ u Γ = g Γ, where the operator S Γ is defined in Equations 22, and { } f i g Γ = R i A i Γ f i A i A i 1 i II f I I he two-level BDDC method is of the form where the preconditioner M 1 = R D,Γ 23 0 A i R i II R D,Γ R Γ M 1 R Γ SΓ RΓ u Γ = M 1 g Γ, A i 1 S R Γ D,Γ has the following form: A i A i 1 0 R i R Γ + ΦS 1 Φ R D,Γ ere Φ is the matrix given by the coarse level basis functions with minimal energy, and it is defined by Φ = R Γ R Γ 0 A i R i II A i A i A i he coarse level problem matrix S is determined by 24 S = N Ri Ai A i I A i A i II A i 1 A i I A i A i I A i 1 A i I A i R i Ri, which is obtained by assembling subdomain matrices; for additional details, cf [3, 7, 5] We know that, under certain assumptions, for any u Γ ŴΓ, 25 u Γ Mu Γ u Γ R Γ S Γ RΓ u Γ C 1 + log/h 2 u Γ Mu Γ his can be established directly by using methods very similar to those of certain studies of the FEI-DP algorithms Denote by E D and P D, the average and jump operators see [10, Formula 64 and 638], on the space W Γ, respectively Central to obtaining the condition number estimate for the preconditioned two-level BDDC operator is a bound for the E D operators see [8, heorem 25] Since E D + P D = I see [10, Lemma 610], we only need to find a bound for the P D operator [10, Lemma 636] gives a bound for the P D operator under the assumptions [10, Assumption 431] for the triangulation and [10, Assumption 6272] for the coefficient ρx of 21 3 A three-level BDDC method For the three-level cases, as in [11], we will not factor the coarse problem matrix S defined in 24 by a direct solver Instead, we will solve the coarse problem approximately by using ideas similar to those for the two-level preconditioners We decompose Ω into N subregions Ω j with diameters Ĥj, j = 1,, N Each subregion Ω j is the union of N j subdomains Ω j i with diameters j i Let Ĥ = max j Ĥj 4

5 and = max i,j j i, for j = 1,, N, and i = 1,, N j We introduce the subregional Schur complement 1 Nj S j = R i Ai A i I A i A i II A i A i I A i A i I A i Ri, and note that the coarse problem matrix S can be assembled from the S j Let e the interface between the subregions; Γ Γ We denote the vector space corresponding to the subdomain edge average variables in Ω i, by W c i Each W c i can be decomposed into a subregion interior part W i c, I b i W c, part W i c,, ie, Wi c = W i c, I b can be further decomposed into a primal subspace W i and a subregion interface, where the subregion interface part Wi c, b Γ c, b and a dual subspace Wi c, b, ie, W i c, = Wi i c, W b c, b We denote the associated product spaces by W c := N Wi c, W c, := N Wi c,, W c, b := N Wi c, b, W c, b := N Wi c, b, and W c, b I := N Wi c, b I Correspondingly, we have W c = W c, b I Wc, b Γ, and W c, b Γ = W c, b Wc, b We denote by Ŵc the subspace of W c of functions that are continuous across Γ We next introduce an interface subspace W c, W c,, for which primal constraints are enforced ere, we only consider edge average constraints We need to change the variables again for all the local coarse matrices corresponding to the edge average constraints he continuous primal subspace is denoted by Ŵc, b he space W c, can be decomposed into W c, = Ŵc, b Wc, b We also denote by F c, b Γ, F c, b Γ, and F c, b Γ, the right hand side spaces corresponding to W c,γ, Ŵc,Γ, and W c,γ, respectively, and will use the same restriction, extension, and scaled restriction operators for F c, b Γ, F Γ, and F c, b Γ We define our three-level preconditioner M 1 by 31 0 A i R i II A i R D,Γ R Γ cf 23, where S 1 A i A i 1 0 R i is an approximation of S 1 R Γ + Φ S 1 Φ R D,Γ, and is defined as follows: given, Ψ 1,,Ψ N,Ψ bγ Ψ F c,, let y = S 1 1 Ψ and ỹ = S Ψ ere Ψ =, y = y 1,,y N,y bγ and ỹ = ỹ 1,,ỹ N,ỹ bγ o solve S y = Ψ by block factorization in the two-level case, we can write 32 S 1 b I S 1 S N b I R 1 S 1 bγ RN S N bγ N S N b Γ R 1 bγ R N bγ i R Γ S i ΓbΓ b 5 R i bγ y 1 bị y N y bγ = Ψ 1 Ψ N Ψ bγ

6 We have 33 y i and N R i S i bγ ΓbΓ b S i = S i 1 b I S i 1 b I Ψ i S i I b R i bγ y, S i i R bγ y bγ = Ψ bγ R i S i bγ S i 1 I b Ψ i In the three-level BDDC algorithm, we need to introduce several restriction, extension, and scaling operators between different subregion spaces he restriction i operator R bγ maps a vector of the space Ŵc, to a vector of the subdomain subspace i Each column of R with a nonzero entry corresponds to an interface node, W i c,γ bγ x Ω i Ω j, shared by the subregion Ω i and certain neighboring subregions bγ i is similar to R bγ which represents the restriction from W c, to W i c, Ri is the b restriction matrix which extracts the subregion part, in the space W i c, b, of the functions in the space W c, b Multiplying each such element of δ i i i x gives us R bd,, R bd, Γ, b and for γ [1/2,, δ i x = ρ γ i x R i R i bγ, R i bγ R i, and R i b with bd, b, respectively ere, we define δ i x as follows: Pj Nx ργ j x, x Ωi Γ, where N x is the set of indices j of the subdomains such that x Ω j and ρ jx is the coefficient of 21 at x in the subregion Ω j In our theory, we assume the ρ i are constant in the subregions Furthermore, R b and R b are the restriction operators from the space W c, b Γ onto its subspace W c, b and W c, b respectively RbΓ : Ŵc, W c, and R bγ : W c, W c, i i are the direct sum of R and R bγ bγ, respectively RbΓ : Ŵc,Γ W c, is the direct sum of R bγ b and the R bγ b he scaled operators R bd, and R bd, b are the direct sums of R i b i i R bd, and R bd, b R bd, is the direct sum of R bγ b and R bd, b RbΓ b Let i = S i ΓbΓ b S i S i 1 I b S i and = diag 1,, N We then introduce a partial assembled Schur complement of S, : W c, F c, by 34 = R bγ R bγ, and define h bγ F c, b Γ, by 35 h bγ = Ψ bγ R i S i bγ S i 1 I b Ψ i he reduced subregion interface problem can be written as: find y bγ Ŵc,, such that 36 R bγ RbΓ y bγ = h bγ When using the three-level preconditioner M 1, we do not solve 36 exactly Instead, we replace y by 37 ỹ bγ = R bd, 1 R bd, h bγ 6

7 We will maintain the same relation between ỹ i and ỹ i 38 ỹ i = S i 1 b I bγ, ie, Ψ i S i I b R i bγ ỹb Γ 4 Some auxiliary results In this section, we will collect a number of results which are needed in our theory In order to avoid a proliferation of constants, we will use the notation A B his means that there are two constants c and C, independent of any parameters, such that ca B CA, where C < and c > 0 For the definition of discrete harmonic functions, see [10, Section 44] Lemma 41 Let D be a cube with vertices A 1 = 0, 0, 0, B 1 =, 0, 0, C 1 =,, 0, D 1 = 0,, 0, A 2 = 0, 0,, B 2 =, 0,, C 2 =,, and D 2 = 0,, with a quasi-uniform triangulation of mesh size h hen, there exists a discrete harmonic function v defined in D such that v A1B 1 1+log h, where v A 1B 1 is the average of v over the edge A 1 B 1, v 2 1 D 1 + log h, and v has a zero average over the other edges Proof: his lemma follows from a result by Brenner and e [1, Lemma 42]: let N be an integer and G N the function defined on 0, 1 by G N x = n=1 1 sin 4n 3πx 4n 3 G N x is even with respect to the midpoint of 0, 1, where it attains its maximum in absolute value Moreover, we have: G N 2 1/2 00 0,1 1 + log N and G N L2 0,1 1; see [1, Lemma 37] Let [, 0] and [0, ] have a mesh inherited from the quasi-uniform meshes on D 1 A 1 and A 1 B 1, respectively, and let g h x be the nodal interpolation of G N x+ 2 hen, we have g h L, 1 + log h, 41 g h 2 1/2 00, 1 + log h and g h L 2, ; see [1, Lemma 37] or [11, Lemma 1] Let τ h x be a function on [0, ] defined as follows: x h 1 0 x h 1, τ h x = 1 h 1 x h 2, x h 2 h 2 x, where h 1 and h 2 are the lengths of the two end intervals hen the following estimates hold: 42 τ h 2 L 2 0, and τ h 2 1/2 00 0, 1 + log h ; see [1, Lemma 36] Define the discrete harmonic function v as 0 on the boundary of D except two open faces A 1 B 1 C 1 D 1 and A 1 B 1 B 2 A 2 It is defined on these two faces by vx 1, x 2, 0 = g h x 2 τ h x 1, for x 1, x 2 A 1 B 1 C 1 D 1, 7

8 vx 1, 0, x 3 = g h x 3 τ h x 1, for x 1, x 3 A 1 B 1 B 2 A 2 It is clear that v A1B log h and that v has a zero average over the other edges Since v is discrete harmonic in D, we have, v 2 1 D = v 2 1/2 D g h 2 1/2, τ h 2 L 2 0, + τ h /2 1 + log, h 00 0, g h 2 L 2, where we have used 41, 42, and [1, Corollary 35] Remark: In Lemma 41, we have constructed the function v for a cube D By using similar ideas, we can construct functions v for other shape-regular polyhedra which will satisfy the same properties and bounds Lemma 42 Let Ω i j be the subdomains in a subregion Ωi, j = 1,, N i, and Vi,j h be the standard continuous piecewise trilinear finite element function space in the subdomain Ω i j with a quasi-uniform fine mesh with mesh size h Denote by E k, k = 1 K j, the edges of the subdomain Ω i j Given the average values of u, ū E k, over each edge, let u Vi,j h h be the discrete Vi,j -harmonic extension in each subdomain Ωi j with the average values given on the edges of Ω i j, j = 1,, N i hen, there exist two positive constants C 1 and C 2, which are independent of Ĥ,, and h, such that C log N i u 2 h 1 Ω i j j=1 N i K j j=1 k 1,k 2=1 ū Ek1 ū Ek2 2 C log N i u 2 h 1 Ω i j j=1 Proof: Without loss of generality, we assume that the subdomains are hexahedral Denote the edges of the subdomain Ω i j by E k, k = 1,, 12, and denote the average values of u over these twelve edges by ū Ek, k = 1,, 12, respectively According to Lemma 41, we can construct eleven discrete harmonic functions φ m, m = 2,, 12, on Ω i j such that and with φ m Ek = { ūem ū E1 1 + log h m = k, 0 m k, 43 φ m 2 1 Ω i j ū E m ū E log, m = 2,, 12 h Let v j = 1 1+log h 12 m=2 φ m + ū E1 ; we then have v j Ek = ū Ek, for k = 1,, 12, 8

9 and 12 v j 2 1 Ω i j = log φ m + ū E1 2 1 Ω i j h m= = 1 + log φ m 2 1 Ω i j h 1 + log φ m 2 1 Ω i j m=2 h m=2 2 1 C 1/ log h 1 + log 12 ū Em ū E1 2 h m=2 1 C log h 12 k=1 ū Ek ū E1 2 ere, we have used 43 for the penultimate inequality By the definition of u, we have, u 2 1 Ω i j v j 2 1 Ω i j 1 C log h 12 k=1 ū Ek ū E1 2 Summing over all the subdomains in the subregion Ω i, we have, C log N i N i 12 u 2 h 1 Ω i j ū Ek ū E1 2 j=1 his proves the first inequality We prove the second inequality as follows: N i 12 j=1 k=1 j=1 k=1 N i 12 ū Ek ū E1 2 = u ū E1 Ek 2 j=1 k=1 N i C 2 1 u ū N i E 1 2 L 2 E k C log h u 2 1 Ω i j j=1 C log N i u 2 h 1 Ω i j j=1 ere, we have used a standard finite element Sobolev inequality, see [10, Lemma 430] for the second inequality and [10, Lemma 416] for the penultimate inequality We complete the proof of the second inequality by using the triangle inequality We now introduce a new mesh on each subregion; we follow [2, 9] he purpose for introducing this new mesh is to relate the quadratic form of Lemma 42 to one for a more conventional finite element space Given a subregion Ω i and subdomains Ω i j, j = 1,, N i, let be a quasi-uniform sub-triangulation of Ω i such that its set of the vertices include the vertices and the midpoints of edges of Ω i j For the hexahedral case, we decomposed each hexahedron into 8 hexahedra by connecting the midpoints of edges We then partition the vertices 9 j=1

10 in the new mesh into two sets he midpoints of edges are called primal and the others are called secondary We call two vertices in the triangulation adjacent if there is an edge of between them, as in the standard finite element context Let U Ω be the continuous piecewise trilinear finite element function space with respect to the new triangulation For a subregion Ω i, U Ω i and U Ω i are defined as restrictions: U Ω i = {u Ω i : u U Ω}, U Ω i = {u Ω i : u U Ω} We define a mapping I Ωi of any function φ, defined at the primal vertices in Ωi, to U Ω i by 44 φx, if x is a primal node; I Ωi φx = the average of the values at all adjacent primal nodes on the edges of Ω i, if x is a vertex of Ω i ; the average of the values at two adjacent primal nodes on the same edge of Ω i, if x is an edge secondary node of Ω i ; the average of the values at all adjacent primal nodes on the boundary of Ω i, if x is a face secondary boundary node of Ω i ; the average of the values at all adjacent primal nodes if x is a interior secondary node of Ω i ; the result of trilinear interpolation using the vertex values, if x is not a vertex of We recall that W c i is the discrete space of the values at the primal nodes given by the subdomain edge average values I Ωi i can be considered as a map from W c to U Ω i or as a map from U Ω i to U Ω i Let I Ωi be the mapping of a function φ defined at the primal vertices on the boundary of Ω i to U Ω i and defined by I Ωiφ = IΩi φ e Ω i, where φ e is any function in W c i such that φ e Ω i = φ he map is well defined since the boundary values of I Ωiφ e only depend on the boundary values of φ e Finally, let I Ωi Ũ Ω i = {ψ = I Ωi φ, φ U Ω i }, Ũ Ω i = {ψ Ω i, ψ ŨΩ i } i also can be considered as a map from W c, to Ũ Ω i Remark: We carefully define the I Ωi and I Ωi so that, if the edge averages of w i W i c, and w j W j c, over an edge E are the same, we have I Ωi w i E = I Ωjw j E ere we need to use a weighted average which has a smaller weight at the two end points But this will not effect our analysis We could also define a weighted edge average of w i and w j and obtain I Ωiw i E = I Ωjw j E for the usual average We list some useful lemmas from [2] For the proofs of Lemma 43 and Lemma 44, see [2, Lemma 61 and Lemma 62], respectively 10

11 Lemma 43 here exists a constant C > 0, independent of and Ω i, the volume of Ω i, such that I Ωi φ 1 Ω i C φ 1 Ω i and I Ωi φ L2 Ω i C φ L2 Ω i, φ U Ω i Lemma 44 For ˆφ Ũ Ω i, inf φ φ U e 1 Ω i ˆφ 1/2 Ω, i Ω i,φ Ω i=ˆφ inf φ φ U e 1 Ω i ˆφ 1/2 Ω i Ω i,φ Ω i=ˆφ Lemma 45 here exist constants C 1 and C 2 > 0, independent of Ĥ,, h, and the coefficient of 21 such that for all w i W i ρ i C 1 I Ωi w i 2 1/2 Ω i 1 + log i w i, w i ρ i C 2 I Ωi w i 2 h 1/2 Ω i, c, b Γ, where i w i, w i = wi i w i = w i 2 and i = S i i ΓbΓ b S i Proof: By the definition of i, we have 1 + log h = inf v W i c v W i c v W i c,v Ω i=w i ρ i inf i w i, w i = 1 + log h 1 + log N i h N i K j ρ i,v Ω i=w i j=1 k 1,k 2=1 inf ρ i I Ω i v 2 1 Ω i,v Ω i=w i ρ i I Ωi w i 1/2 Ω i j=1 v Ek1 v Ek2 2 v W i c inf u V h i,j,ūe=v,e Ωi j S i 1 b I S i inf v 2 S i,v Ω i=w i u 2 1 Ω i j inf φ e U Ω i,φ Ω i=i Ωi wi ρ i φ 2 1 Ω i We use Lemma 42 for the third bound, the definitions of I Ωi and fifth bounds, and Lemma 44 for the final one and I Ωi for the fourth o be fully rigorous, we assume that there is a quasi-uniform coarse triangulation of each subregion We can then obtain uniform constants C 1 and C 2 in Lemma 45, which work for all the subregions We define the interface averages operator Ê D b on W c, as Ê D b = RbΓ R bd,,which computes the averages across the subregion interface Γ and then distributes the averages to the boundary points of the subregions he interface average operator Ê D b has the following property: Lemma 46 Ê b D w bγ 2 e C 1 + log Ĥ 11 2 w bγ 2 e,

12 for any w bγ W c, b Γ, where C is a positive constant independent of Ĥ,, h, and the coefficients of 21 ere is define in 34 i Proof: Let w i = R bγ w bγ W i c, We rewrite the formula for v := w Êb Γ w bγ for an arbitrary element w bγ W c,, and find that for i = 1,, N, 45 v i x := w bγ x Êb Γ w bγ x i = δ j w ix w j x, x Ω i Γ j N x ere N x is the set of indices of the subregions that have x on their boundaries We have Ê D bw bγ 2 e = w i v i 2 2 w i i v i i 2 and w i bγ 2 e = w i 2 i We can therefore focus on the estimate of the contribution from a single subregion Ω i and proceed as in the proof of [10, Lemma 636] We will also use the simple inequality 46 ρ i δ 2 j By Lemma 45, minρ i, ρ j, for γ [1/2, 47 i 1 v i, v i C log h ρ i I Ω i v i 2 1/2 Ω i Let L i = I Ωiv i We have, by using a partition of unity as in [10, Lemma 636], L i = I θ F L i + I θ E L i + θ V L i V, F Ω i E Ω i V Ω i where I is the nodal piecewise linear interpolant on the coarse mesh We note that the analysis for face and edge terms is almost identical to that in [10, Lemma 636] But the vertex terms are different because of I Ωi We only need to consider the vertex term when two subregion share at least an edge his make the analysis simpler than in the proof of [10, Lemma 636] Face erms First consider, 48 I θ F L i = I θ F I Ωi δ j w i w j Similar to [10, Lemma 636], we obtain, by using 46, ρ i I θ F I Ωi δ j w i w j 2 1/2 Ω i = ρ i δ 2 j I θ F I Ωi w i w j 2 1/2 Ω i minρ i, ρ j I θ F I Ωi I Ωi 3 minρ i, ρ j w i I Ωi w i F I Ωiw j F 2 1/2 Ω i w i F I Ωi w j I Ωiw j F + I θ F I Ωi w i I Ωiw i F 2 1/2 Ω i + I θ F I Ωi w j I Ωiw j F 2 1/2 Ω i + θ F I Ωiw i F I Ωiw j F 2 1/2 Ω i 12

13 By the definition of I Ωi, I θ F I Ωi w j = I θ F I Ωj w j and I Ωi w j F = I Ωj w j F By [10, Lemma 426], the first and second terms in 48 can be estimated as follows: minρ i, ρ j I θ F I Ωi w i I Ωiw i F 2 1/2 Ω i + I θ F I Ωi w j I Ωiw j F 2 1/2 Ω i = minρ i, ρ j I θ F I Ωi w i I Ωiw i F 2 1/2 Ω i + I θ F I Ωj w j I Ωjw j F 2 1/2 Ω i 2 C 1 + log Ĥ ρ i I Ωi w i 2 1/2 Ω i + ρ j I Ωj w j 2 1/2 Ω j Let E F Since the edge averages of w i and w j are the same, by the definition of I Ωi and I Ωj, we have I Ωi w i E = I Ωjw j E As we have pointed out before, we use the weighted average which has a smaller weight at the two end points We then have 49 I Ωi 2 I Ωi w i F I Ωj w j F 2 w i E I Ωi w i F 2 + I Ωjw j E I Ωjw j F 2 It is sufficient to consider the first term on the right hand side Using [10, Lemma 430], we find I Ωiw i E I Ωiw i F 2 = I Ωi w i I Ωi w i F E 2 C/Ĥi I Ωi w i I Ωiw i F 2 L 2 E, and, by using [10, Lemma 417] and the Poincaré inequality given as [10, Lemma A17], I Ωiw i E I Ωiw i F 2 C/Ĥ i 1 + log Ĥ I Ωi w i I Ωiw i F 2 1/2 F Combining this with the bound for θ F in [10, Lemma 426], we have: minρ i, ρ j θ F I Ωi log Ĥ C w i F I Ωi w j F 2 1/2 Ω i ρ i I Ωi w i 2 1/2 Ω i + ρ j I Ωj w j 2 1/2 Ω j Edge erms We can develop the same estimate as in [10, Lemma 634] For simplicity, we only consider an edge E common to four subregions Ω i, Ω j, Ω k, and Ω l 410 ρ i I θ E L i 2 1/2 Ω i ρ i I θ E I Ωi δ j w i w j 2 1/2 Ω i + I θ E I Ωi δ k w i w k 2 1/2 Ω i + I θ E I Ωi δ l w i w l 2 1/2 Ω i 13

14 We recall that δ j, δ k, and δ l are constants By the definition of I Ωi θ E I Ωi and, w j = θ E I Ωj I Ωi, I Ωj w j, θ E I Ωi w i E = I Ωj, I Ωk, and I Ωl, we have w k = θ E I Ωk w k, θ E I Ωi w j E = I Ωkw k E = I Ωlw l E w l = θ E I Ωl w l, We assume that Ω i shares a face with Ω j as well as Ω l, and shares an edge only with Ω k We consider the second term in 410 first By [10, Lemmas 419 and 417], and 46, we have ρ i I θ E I Ωi δ k w i w k 2 1/2 Ω i Cρ i δ 2 k I θ E I Ωi 2C w i I Ωi ρ i I θ E I Ωi w i I Ωi ρ k I θ E I Ωk 2C ρ i I Ωi w i I Ωi 2C 1 + log Ĥ 2C 1 + log Ĥ w i E θ E I Ωk w i E 2 L 2 E + w k I Ωkw k E 2 L 2 E w i E 2 L 2 E + ρ k I Ωk w k I Ωkw k E 2 L 2 E w k I Ωkw k E 2 L 2 E ρ i I Ωi w i 2 1/2 F i + ρ k I Ωk w k 2 1/2 F k ρ i I Ωi w i 2 1/2 Ω i + ρ k I Ωk w k 2 1/2 Ω k, where F i is a face of Ω i and F k is a face of Ω k, and F i and F k share the edge E he first term and the third term can be estimated similarly Vertex erms We can do the estimate similarly to that of the proof in [10, Lemma 636] We have 411 ρ i θ V L i V 2 1/2 Ω i = ρ i θ V I Ωi v i V 2 1/2 Ω i By 45 and the definition of I Ωi, we see that I Ωi v iv is nonzero only when two subregions share one or several edges with a common vertex V In the definition of I Ωi, we denote by E i,m, m = 1, 2, 3, the edges in Ω i which share V Denote by v i,m the primal nodes on the edges E i,m which are adjacent to V By the definition of I Ωi, 411, and θ V 2 1/2 Ω i C i, we have, 412 ρ i θ V I Ωi v iv 2 1/2 Ω i Cρ i v i v i,m 2 θ V 2 1/2 Ω i m Cρ i i v i v i,m 2 m Let us look at the first term in 412, the other terms can be estimated in the same way 14

15 ρ i i v i v i,1 2 = ρ i i δ j w iv i,1 w j v i,1 2 j,e i,1 Ω j C minρ i, ρ j i w i v i,1 w j v i,1 2 j,e i,1 Ω j = C minρ i, ρ j i I Ω i w i v i,1 I Ωj w j v i,1 2 j,e i,1 Ω j C minρ i, ρ j i I Ωi w i v i,1 I Ωiw i Ei,1 2 + j,e i,1 Ω j I Ωj w jv i,1 I Ωjw j Ei,1 2 C minρ i, ρ j i j,e i,1 Ω j I Ωj j,ei,1 Ω j I Ωi w i I Ωiw i Ei,1 v i,1 2 + i w j I Ωjw j Ei,1 v i,1 2 C minρ i, ρ j I Ωi w i I Ωiw i Ei,1 2 L 2 E + i,1 j,ei,1 Ω j I Ωj w j I Ωjw j Ei,1 2 L 2 E i,1 C 1 + log Ĥ ρ i I Ωi w i 2 1/2 Ω i + ρ j I Ωj w j 2 1/2 Ω i For the third equality, we use here that v i,1 is a primary node For the fourth inequality, we use that I Ωiw i Ei,1 = I Ωjw j Ei,1 We use [10, Lemmas B5] for the sixth inequality and [10, Lemma 417] for the last inequality Combining all face, edge, and vertex terms, we obtain ρ i I Ωi v i 2 1/2 Ω i C 1 + log Ĥ ρ j I Ωi w j 2 1/2 Ω j j: Ω j Ω i Using 413, Lemma 45 and 47, we obtain i v i, v i = v i 2 1 C i log h ρ i I Ω i v i 2 1/2 Ω i log Ĥ CC log h ρ j I Ωi w j 2 1/2 Ω j j: Ω j Ω i 2 C C log Ĥ j w j, w j C 1 = C C 2 C log Ĥ j: Ω j Ω i 2 j: Ω j Ω i 15 w j 2 j

16 Lemma 47 Given any u Γ ŴΓ, let Ψ = Φ RD,Γ u Γ We have, Ψ S 1 Ψ Ψ S 1 Ψ C 1 + log Ĥ 2 Ψ S 1 Ψ 414 = = Proof: Using 33, 35, and 36, we have N Ψ S 1 Ψ = Ψ i y i + Ψ y bγ Ψ i Ψ i S i 1 I b Ψ i S i 1 I b Ψ i + h y bγ = S i I b R i bγ y + h bγ + Ψ i Using 38, 35, and 37, we also have = = Ψ S 1 Ψ = N Ψ i Ψ i S i 1 I b Ψ i Ψ i ỹ i + Ψ ỹ bγ S i 1 I b Ψ i + h ỹ bγ = S i 1 I b Ψ i S i I b R i bγ ỹb Γ + h bγ + Ψ i S i 1 I b Ψ i R i S i bγ ΓI b S i 1 I b Ψ i y bγ + h 1 R bγ RbΓ h bγ R i S i bγ S i 1 I b Ψ i ỹ bγ + h R bd, 1 R bd, h bγ We only need to compare h 1 R bγ RbΓ h bγ and h R bd, 1 R bd, h bγ for any h bγ F c, b Γ he following estimate is established as [6, heorem 1] Let w bγ = R bγ RbΓ h bγ Ŵc, and v = 1 R bd, h bγ W c, Noting the fact that R bγ R bd, = R bd, Γ R b bγ = I and using 415, we have, h 1 R bγ RbΓ h bγ = h w bγ = h R bd, Γ R b bγ w bγ = h R bd, 1 RbΓ w bγ = 1 R bd, h bγ RbΓ w bγ = v RbΓ w bγ =< v bγ, RbΓ w bγ > e < v bγ,v bγ > 1/2 e < RbΓ w bγ, RbΓ w bγ > 1/2 e = h R bd, 1 R 1/2 bd, h bγ h 1 1/2 R bγ RbΓ h bγ 16

17 We obtain that On the other hand, h 1 R bγ RbΓ h bγ h R bd, 1 R bd, h bγ h R bd, 1 R bd, h bγ = w R bγ R RbD, 1 R bd, h bγ = < w bγ, R bd, 1 R bd,bγ h bγ > b er bγ eb «=< w Rb e bγ, R bd, v > b er Γ bγ eb «Rb e Γ < w bγ,w bγ > 1/2 ber «< R bγ eb bd, Γv Rb e b bγ, R Γ = bd, b Γv bγ > 1/2 ber bγ eb «Rb e Γ 1 1/2 h R bγ RbΓ h bγ < RbΓ R bd, Γv b bγ, RbΓ R bd, Γv b bγ > 1/2 e = h 1 1/2 R bγ RbΓ h bγ Ê D bv bγ e C 1 + log Ĥ h 1 1/2 b Γ R bγ RbΓ h bγ v bγ e = C 1 + log Ĥ h 1 1/2 b Γ R bγ RbΓ h bγ h R bd, 1 R 1/2 bd, h bγ, where we use Lemma 46 for the penultimate inequality Finally we obtain that 2 h R bd, 1 R bd, h bγ C 1 + log Ĥ h 1 b Γ R bγ RbΓ h bγ 5 Condition number estimate for the new preconditioner In order to estimate the condition number for the system with the new preconditioner M 1, we compare it to the system with the preconditioner M 1 Lemma 51 Given any u Γ ŴΓ, 51 u Γ M 1 u Γ u Γ M 1 u Γ C Proof: We have, for any u Γ ŴΓ, 1 + log Ĥ 2 u Γ M 1 u Γ u Γ M 1 u Γ = u R Γ D,Γ R Γ + u Γ R D,Γ ΦS 1 Φ RD,Γ u Γ 0 A i R i II A i 17 A i A i 1 0 R i R Γ R D,Γ u Γ

18 and u M Γ 1 u Γ = u R Γ D,Γ R Γ + u R Γ D,Γ 1 Φ S Φ RD,Γ u Γ 0 A i R i II A i A i A i 1 0 R i R Γ R D,Γ u Γ We obtain our result by using Lemma 47 heorem 52 he condition number for the system with the three-level preconditioner M 1 is bounded by C1 + log Ĥ log h 2 Proof: Combining the condition number bound, given in 25, for the two-level BDDC method, and Lemma 51, we find that the condition number for the three-level method is bounded by C1 + log Ĥ log h 2 6 Using Chebyshev iterations Another approach to the three-level BDDC methods is to use an iterative method with a preconditioner to solve 36 ere, we consider a Chebyshev method with a fixed number of iterations and use R bd, 1 R bd, as a preconditioner Denoting the eigenvalues of R bd, 1 R bd, R bγ Rby λ j, we need two input parameters l and u, estimates for the minimum and maximum values of λ j, for the Chebyshev iterations From our analysis above, we know that l = 1 and max j λ j C1+log Ĥ 2 1+log h 2 We can use the conjugate gradient method to obtain an estimate for the largest eigenvalue at the beginning of the computation to choose a proper u Let α = 2 l+u, µ = u+l u l, and σ j = 1 αλ j As for the two dimensional case in [11, Section 6], we have the following theorem No new ideas are required 1 heorem 61 he condition number using the three-level preconditioner M with k Chebyshev iterations is bounded by C C2k C 1 + log 1k h 2, where and C2k C 1k 1 as k C 1 k = min 1 coshk cosh 1 µσ j j coshk cosh 1 µ C 2 k = max 1 coshk cosh 1 µσ j j coshk cosh 1, µ 7 Numerical experiments We have applied our two three-level BDDC algorithms to the model problem 21, where Ω = [0, 1] 3 We decompose the unit cube into N N N subregions with the side-length Ĥ = 1/ N and each subregion into N N N subdomains with the side-length = Ĥ/N Equation 21 is discretized, in each subdomain, by conforming piecewise trilinear elements with an element diameter h he preconditioned conjugate gradient iteration is stopped when the norm of the residual has been reduced by a factor of

19 able 1 Eigenvalue bounds and iteration counts with the preconditioner M f 1 with a change of the Ĥ number of subregions, = 3 and h = 3 Case 1 Case 2 Num of Subregions Iter Cond # Iter Cond # able 2 Eigenvalue bounds and iteration counts with the preconditioner fm 1 with a change of the number of subdomains and the size of subdomain problems with subregions Case 1 Case 2 Case 1 Case 2 Ĥ Iter Cond # Iter Cond # h Iter Cond # Iter Cond # We have carried out two different sets of experiments to obtain iteration counts and condition number estimates All the experimental results are fully consistent with our theory In the first set of experiments, we use the first preconditioner M 1 We take the coefficient ρ 1 in Case 1 In Case 2, ρ is constant in one direction with a checkerboard pattern in the cross sections, where we take ρ = 1 or ρ = 100 he coefficients in both cases satisfy [10, Assumption 6272], ie, for all pairs of subdomains which have a vertex but not an edge in common, there exists an acceptable edge path see [10, Definition 626] between these two subdomains able 1 gives the iteration counts and condition number estimates with a change of the number of subregions We find that the condition numbers are independent of the number of subregions able 2 gives results with a change of the number of subdomains and the size of the subdomain problems In the second set of experiments, we use the second preconditioner M 1 and take the coefficient ρ 1 We use the Preconditioned Conjugate Gradient PCG to esti- mate the largest eigenvalue of R bd, 1 R bd, R bγ, RbΓ which is approximately For subdomains and h = 3, we have a condition number estimate of for the two-level preconditioned BDDC operator We select different values of u, the upper bound eigenvalue estimate of the preconditioned system, and k to see how the condition number changes We take u = 23 and u = 3 in able 3 and 4, respectively We also evaluate C 1 k for k = 1, 2, 3, 4, 5 From these two tables, we find that the smallest eigenvalue is bounded from below by C 1 k and the condition number estimate becomes closer to 18767, the value for the two-level case, as k increases We also see that if we can get more precise estimate for the largest R eigenvalue of bd, 1 R bd, R bγ, RbΓ we need fewer Chebyshev iterations to 19

20 able 3 Eigenvalue bounds and iteration counts with the preconditioner M c 1, u = 23, Ĥ subregions, = 6 and h = 3 k Iter C 1 k λ min λ max Cond # able 4 Eigenvalue bounds and iteration counts with the preconditioner M c 1, u = 3, subregions, Ĥ = 6 and h = 3 k Iter C1k λmin λmax Cond # get a condition number, close to that of the two-level case owever, the iteration count is not very sensitive to the choice of u Acknowledgments he author is grateful to Professor Olof Widlund for suggesting this problem and all the help and time he has devoted to my work REFERENCES [1] Susanne C Brenner and Qingmi e Lower bounds for three-dimensional nonoverlapping domain decomposition algorithms Numer Math, 93: , 2003 [2] Lawrence C Cowsar, Jan Mandel, and Mary F Wheeler Balancing domain decomposition for mixed finite elelements Math Comp, 64: , 1995 [3] Clark R Dohrmann A preconditioner for substructuring based on constrained energy minimization SIAM J Sci Comput, 251: , 2003 [4] Axel Klawonn and Olof B Widlund Dual-Primal FEI methods for linear elasticity echnical Report R-855, Department of Computer Science, New York University, 2004 [5] Jing Li and Olof B Widlund FEI-DP, BDDC and block Cholesky methods echnical Report R-857, Department of Computer Science, New York University, 2004 [6] Jing Li and Olof B Widlund BDDC algorithms for incompressible Stokes equations echnical Report R-861, Department of Computer Science, New York University, 2005 [7] Jan Mandel and Clark R Dohrmann Convergence of a balancing domain decomposition by constraints and energy minimization Numer Linear Algebra Appl, 107: , 2003 [8] Jan Mandel, Clark R Dohrmann, and Radek ezaur An algebraic theory for primal and dual substructuring methods by constraints Appl Numer Math, 2004 o appear [9] Marcus V Sarkis Nonstandard coarse spaces and Schwarz methods for elliptic problems with discontinuous coefficients using non-conforming elements Numer Math, 773: , 1997 [10] Andrea oselli and Olof B Widlund Domain Decomposition Methods - Algorithms and heory, volume 34 of Springer Series in Computational Mathematics Springer Verlag, Berlin- eidelberg-new York, 2004 [11] Xuemin u hree-level BDDC in two dimensions echnical Report R , Department of Computer Science, New York University,