An Explicit Harmonic Extension for The Constant-like Basis And Its Application to Domain Decompositions 1

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1 An Explicit Harmonic Extension or The Constant-like Basis And Its Application to Domain Decompositions 1 Qiya Hu and Dudu Sun LSEC, Institute o Computational Mathematics and Scientiic/Engineering Computing, Academy o Mathematics and Systems Science, Chinese Academy o Sciences, Abstract In this paper we are concerned with substructuring methods or the second-order elliptic problems in three-dimensional domains. We irst design a simple and completely explicit nearly harmonic extension or the constant-like basis unction (i.e., the ace basis unction), and then deine a coarse subspace based on this nearly harmonic extension. Own to the resulting coarse solver, we develop a kind o substructuring preconditioner with inexact solvers. We show that the condition number o the preconditioned system grows only as the logarithm o the dimension o the local problem associated with an individual substructure, and is independent o possible jumps o the coeicient in the elliptic equation. Numerical experiments conirms the theoretical results. Key words. domain decomposition, coarse subspace, constant-like basis, nearly harmonic extension, preconditioner, inexact solvers, condition number AMS(MOS) subject classiication 65F10, 65N30, 65N55 1 Introduction Non-overlapping domain decomposition methods (DDMs) have been shown to be powerul techniques or solving partial dierential equations, especially or the case with large jump coeicients. One s main task in non-overlapping DDMs is the construction o an eicient substructuring preconditioner or discretization system associated with the underlying partial dierential equations. The construction o this preconditioner has been investigated rom various ways and to various models in literature, see, or example, [1]-[11], [13]-[18], [20]-[24], [26]-[27], [29]. Most non-overlapping DDMs studied so ar require exact subdomain solvers; we reer [28] and [32] (and the reerences cited therein). Such a requirement severely degrade the eiciency o the methods. There are only a ew works studying substructuring methods with inexact subdomain solvers [3], [4], [11] and [6]. The essential diiculty is that discrete harmonic extensions on each subdomain are used in non-overlapping domain decomposition methods. In [3], analysis and numerical experiments with inexact algorithms o Neumann- Dirichlet type was done under the additional assumption o high accuracy o the inexact solvers. In [4], the harmonic extension on a subdomain was replaced by a simple average extension, and substructuring preconditioners with the average extension are constructed. Because o such average extension, nearly optimal convergence can not be gotten or these substructuring preconditioners. To avoid harmonic extensions, [11] considered so called approximate harmonic basis unctions, which still involve high accuracy o the inexact solvers. Another way to construct substructuring preconditioner with inexact solvers was considered in [6] (mainly or two dimensions). In this preconditioner, overlapping ace subspaces are used, and harmonic extension is used only in the deinition o coarse subspace. How to design nearly harmonic extensions involved in the coarse subspace is the core problem in 1 The work was supported by The Key Project o Natural Science Foundation o China G , National Basic Research Program o China G2005CB and Natural Science Foundation o China G ( hqy@lsec.cc.ac.cn) 1

2 the construction o such substructuring preconditioner. In the case with three dimensions, the coarse subspace may consist o the edge basis and ace basis, which are constant-like unctions. Then, one needs only to deine suitable extensions or constant-like unctions on aces. Such extensions, which depend on the geometric shapes o the subdomains, was irst studied in [8]. In the present paper we introduce a new variant o substructuring preconditioner with inexact solvers considered in [6] (see also [32]). The main contribution o the paper is the design o a simple and completely explicit extension or the constant-like basis unction. This extension plays a key role in the substructuring preconditioner with inexact solvers. We show that the new substructuring preconditioner possesses nearly optimal convergence, which is independent o possible large jumps o the coeicient across the interace. For the new method, no additional assumption is required. The outline o the remainder o the paper is as ollows. In Section 2, we introduce some notation and our motive. In Section 3, we present an explicit extension o the constantlike unction. The results on the substructuring preconditioner with inexact solvers are described in Section 4. In Section 5, we prove the stability o the extension o constant-like unction, which is used in Section 4. Some numerical results are reported in Section 6. 2 Preliminaries 2.1 Domain decomposition Let Ω be a bounded polyhedron in R 3. Consider the model problem { div(ω u) =, in Ω, u = 0, on Ω, (2.1) where ω L (Ω) is a positive unction. Let H0 1(Ω) denote the standard Sobolev space, and let (, ) denote the L2 (Ω)-inner product. The weak ormulation o (2.1) in H0 1 (Ω) is then given by the ollowing. Find u H0 1 (Ω) such that where (, ) is the scalar product in L 2 (Ω), and A(u, v) = ω u vdp. A(u, v) = (, v) v H 1 0 (Ω), (2.2) Ω We will apply a kind o non-overlapping domain decomposition method to solving (2.2). For simplicity o exposition, we consider only the case with matching grids in this paper. Let T h = {τ i } be a regular and quasi-uniorm triangulation o Ω with τ i s being nonoverlapping simplexes o size h ( (0, 1]). The set o notes o T h is denoted by N h. We then deine V h (Ω) to be the piecewise linear inite element subspace o H0 1 (Ω) associated with T h : V h (Ω) = {v H0 1 (Ω) : v τ P 1 τ T h }, where P 1 is the space o linear polynomials. Then the inite element approximation or (2.2) is to ind u h V h (Ω) such that A(u h, v h ) = (, v h ), v h V h (Ω). (2.3) Let Ω be decomposed into the union o n polyhedrons Ω 1,, Ωn, which satisy Ω i Ω j = when i j. We assume that each Ω k can be written as a union o boundaries o 2

3 elements in T h, and all Ω k are o size H in the usual sense (see [5] and [32]). Without loss o generality, we assume that the coeicient ω(p) is piecewise constant, then each subdomain Ω k is chosen such that ω(p) equals to a constant ω k in Ω k. Note that {Ω k } may not constitute a triangulation o Ω. The common part o two neighboring subdomains Ω i and Ω j may be a vertex, an edge or a ace. In particular, we denote by Γ ij the common ace o two neighboring subdomains Ω i and Ω j (i.e., Γ ij = Ω i Ω j ). The union o all Γ ij is denoted by Γ, which is called the interace. In this paper, we choose Dirichlet data as the interace unknown. Deine the operator A h : V h (Ω) V h (Ω) by (A h v, w) = A(v, w) = N ω k v wdx, v V h (Ω), w V h (Ω). Ω k k=1 The equation (2.3) can be written as A h u h = h, u h V h (Ω). (2.4) The goal o this paper is to construct a substructuring preconditioner or A h based on the domain decomposition described above. 2.2 Notations To introduce the new method, we need some more notations. Throughout this paper, a subset G o Ω are always understood as an open set. subdomain spaces For subdomain Ω k, deine and Set and deine V h (Ω k ) = {v Ωk : v V h (Ω)}, V p h (Ω k) = {v h V h (Ω) : supp v h Ω k }. Ω ij = Ω i Γ ij Ω j, V p h (Ω ij) = {v h V h (Ω) : supp v h Ω ij }. interace space and ace spaces As usual, we deine the (global) interace space by W h (Γ) = {v Γ : v V h (Ω)}. For each Ω k, set For a ace = Γ ij, deine W h ( Ω k ) = {v Ωk : v W h (Γ)}. W h () = {φ h W h (Γ) : supp φ h }. interpolation-type operator and constant-like basis For a subset G o Γ, deine the interpolation-type operator I 0 G : W h(γ) W h (Γ) as (I 0 Gφ h )(p) = { φh (p), i p N h G, 0, i p N h (Γ\G). 3

4 In particular, we have (I 0 G1)(p) = { 1, i p Nh G, 0, i p N h (Γ\G). I G is an edge or a ace generated by the domain decomposition, we call φ G = IG 0 1 to be constant-like basis unction on G, which will be used repeatedly. integration average and algebraic average For a unction ϕ h W h (Γ), let γ G (ϕ h ) denote the integration average o ϕ h on G, and let γ h,g (ϕ) denote the algebraic average o the values o ϕ on the nodes in G. sets o aces, edges, vertices and subdomains For convergence, let F Γ denote the set o all the aces Γ ij. Besides, let E Γ and V Γ denote the set o the interior edges and the set o interior vertices generated by the decomposition Ω = Ωk, respectively. For an edge e E Γ, let Qe denote the set o the indices k o the subdomains Ω k which contain e as an edge. Namely, Deine Qe = {k : e Ω k }. Ωe = k Q e Ω k, e E Γ. ace inner-products, scaling norm and interace norm For a subset G o Γ, let, G denote the L 2 inner product on G. In particular, the, Γ is abbreviated as,. Let 0, G denote the norm induced rom, G. For a sub-aces G o Γ, let H G denote the size o G. Deine the scaled norm For convenience, deine φ 1 2, G = ( φ 2 1 2, G + H 1 G φ 2 0, G) 1 1 2, φ H 2 (G). n φ h,γ = ( k=1 ω k φ h 2 1 2, Ω k ) 1 2, φh W h (Γ). discrete norms Discrete norms (or semi-norms) o inite element unctions will be used repeatedly in this paper, since the discrete norms are deined on a set o nodes only, and do not depend on the geometric shape o the underlying domain. We irst give deinitions o two well known discrete norms (reer to [32]), which are equivalent to their respective continuous norms. For v h V h (Ω k ), the discrete H 1 seminorm is deined by v h 2 1, h, Ω k = h 3 p i, p j N h Ω k v h (p i ) v h (p j ) 2, where p i and p j denote two neighboring nodes. Similarly, the discrete L 2 norm on an edge e o Ω k is deined by v h 2 0, h, e = h v h (p) 2. H norms p N h e 4

5 For ϕ h W h () with = Γ ij, deine ϕ h 2 1 = ϕ h 2 1 H 200 () 2, + ϕ h (x) 2 dist(x, ) ds(x). Hereater, dist(x, ) denotes the shortest distance rom a point x to the boundary. It is known that ϕ h 2 = 1 H 2 ϕ h 2 = 1 00 () 2, Ω i ϕ h 2 1 2, Ω, j where ϕ h W h (Γ) denotes the zero extension o ϕ h. Moreover, we have ϕ h 2 H () = The corresponding discrete semi-norm is deined by ϕ h 2 1 = h 2 h, H 2 00 () ϕ h (x) 2 dist(x, ) ds(x). p N h ϕ h (p) 2 dist(p, ). spectrally equivalences For simplicity, we will requently use the notations and =. For any two non-negative quantities x and y, x y means that x Cy or some constant C independent o mesh size h, subdomain size d and the related parameters. x = y means x y and y x. 2.3 Motivation We irst recall the main ideas o the existing substucturing preconditioners. Let E k : W h ( Ω k ) V h (Ω k ) be the discrete harmonic extension. Deine the harmonic subspace V Γ h (Ω) = {v h V h (Ω) : v h Ωk = E k (φ h Ωk ) (k = 1,, n) or some φ h W h (Γ)}. Then, we have the initial space decomposition V h (Ω) = n V p h (Ω k) + Vh Γ (Ω). k=1 Let A h,k : V p h (Ω k) V p h (Ω k) be the restriction o the operator A h on the local space V p h (Ω k), and let B h,γ : Vh Γ(Ω) V h Γ (Ω) be a symmetric and positive deinite operator which is spectrally equivalent to the restriction o the operator A h on the harmonic subspace Vh Γ (Ω). Then, the classical substructuring preconditioner (reer to [5]) can be deined in the rough orm N Bold 1 = A 1 h,k Q k + Bh,Γ 1 Q Γ, (2.5) k=1 where Q k and Q Γ denote the standard L 2 projectors into their respective subspaces. For the preconditioner B old, we have (see [5]) cond(b 1 old A h) log 2 (H/h). (2.6) In many applications, the subspaces V p h (Ω k) still have high dimensions, so it is expensive to use the exact solvers A 1 h,k. 5

6 It was shown in [3] that substructuring preconditioners with inexact solvers Bh,k 1 still possess nearly optimal convergence, i each B h,k has some spectrally approximation to A h,k (the usual spectrally equivalence is not enough). Hereater, inexact means that B h,k is only spectrally equivalent to A h,k, or example, B h,k is a multigrid preconditioner or A h,k. It seems diicult to design an eicient substructuring preconditioner with completely inexact solvers Bh,k 1, instead o A 1 h,k itsel or its approximation. In essence, one has to modiy the harmonic subspace Vh Γ(Ω) by replacing each harmonic extension E k with another extension. In [4], a substructuring preconditioner Bbpv 1 with inexact solvers was been designed by replacing each harmonic extension E k with a simple average extension. It has been shown that the condition number o the resulting preconditioned system can be estimated by cond(b 1 bpv A h) H/h. (2.7) In [11], another substructuring preconditioner Bhlm 1 with inexact solvers was been designed by replacing each harmonic extension E k with an approximate harmonic extension. I the approximate harmonic extension is exact enough, then cond(b 1 blm A h) log 2 (H/h). (2.8) The approximate harmonic extension can be deined by approximate harmonic basis unctions. It is not practical to compute all the approximate harmonic basis unctions. Because o this, an alternative method, which still require high accuracy o B h,k, was considered in [11]. Another way to construct substructuring preconditioner with inexact solvers was considered in [6] (mainly or two dimensions). Let Ŵ h 0(Γ) be a subspace o W h(γ), such that a unction ϕ Ŵ h 0 (Γ) equals a constant C at every nodes in each ace o Γ. Deine the coarse subspace ˆV 0 h (Ω) = {v h V h (Ω) : v h Ωk = E k (φ h Ωk ) (k = 1,, n) or some φ h Ŵ 0 h (Γ)}. Then, we have the space decomposition V h (Ω) = ˆV 0 h (Ω) + Γ ij V p h (Ω ij). Let B 0 : ˆV 0 h (Ω) ˆV h 0(Ω) and B ij : V p h (Ω ij) V p h (Ω ij) be symmetric and positive deinite operators which are spectrally equivalent to the restrictions o the operator A h on the coarse subspace ˆV h 0(Ω) and the local subspace V p h (Ω ij), respectively. Then, we can deine a substructuring preconditioner as (reer to [6] and [32]) B 1 dw = B 1 0 Q 0 + Γ ij B 1 ij Q ij, (2.9) where Q 0 and Q ij denote the standard L 2 projectors into their respective subspaces. For the preconditioner B dw, we have (reer to [6] and [32]) cond(b 1 dw A h) log 2 (H/h). (2.10) As pointed out in [32], the advantage o the preconditioner B dw is that one can use inexact solver B ij or the restriction o A h on V p h (Ω ij), but an inexact solver B0 1 is hard to come by as harmonic extensions in the deinition o ˆV h 0 (Ω) mean an exact solver. As an alternate method, one can use a nearly harmonic extension to replace the exact harmonic extension E k. However, the design o such approximate harmonic extension is also diicult 6

7 or the general case that either the subdomain Ω k is a general polyhedron or T h is a general quasi-uniorm triangulation (an attempt or two dimension is given in [19]). From the deinition o the coarse subspace ˆV h 0 (Ω), we know that one needs only to design such extension or unctions in the subspace Ŵ h 0 (Γ). In essence, one needs only to deine a suitable extension or the constant-like basis unction on each ace o Γ. This kind o extension was irst studied in [8]. For the purpose o applications, the deinition o such extension would be suiciently simple, otherwise, the action o B0 1 is diicult to implement. In this paper, we construct a completely explicit extension or the constant-like basis unction on each ace o Γ. As we will see, the implementation o the new extensions is very convenient and cheap. We can deine a new variant o the coarse subspace ˆV h 0 (Ω) based on the extensions. Then, we use the new coarse subspace to construct a substructuring preconditioner with inexact solvers. 2.4 Basic tools In the analysis later, we will use some basic estimates on H 1 2 norms. The ollowing results are well-known (see, or example, [32]). Lemma 2.1 Let e and be an edge and a ace o Ω k. Then, and φ h 0, log 1 2 (H/h) φh 1 2, Ω k, φ h W h ( Ω k ), (2.11) Ieφ 0 h 1 2, Ω k log 1 2 (H/h) φh 1 2, Ω, φ k h W h ( Ω k ) (2.12) I 0 φ h 1 H 2 log(d/h) φ h 1 00 () 2, Ω, φ k h W h ( Ω k ). (2.13) The ollowing result can be proved as in Lemma 6.2 in [18], together with the standard technique. Lemma 2.2 Let e be an edge o Ω k. Then, φ h γ h,e(φ h ) 1 2, Ω k log 1 2 (H/h) φh 1 2, Ω, φ k h W h ( Ω k ). (2.14) 3 Extensions or particular unctions As in Subsection 2.3, the desired coarse subspace o V h (Ω) is always associated with an interace coarse subspace. Let I 01 and I0 e1 denote the constant-like basis unctions on the ace and the coarse edge e, respectively (see Subsection 2.2). In this paper, we consider the interace coarse subspace W 0 h (Γ) = span{i 0 F 1, I 0 e1, ϕ p : F F Γ, e E Γ, p V Γ }. Hereater, ϕ p denotes the nodal basis unction on the node p N h Ω. In this section, we construct an approximation or the restriction o the discrete harmonic extension E k on W 0 h (Γ). For convenience, let denote a general ace Γ ij, and let φ = I 0 1 W h (F ) be the constant-like basis unction on. 7

8 3.1 An explicit extension o φ In this subsection, we deine a stable extension Eφ V p h (Ω ij), which satisies (Eφ)(p) = φ(p) when p. For a node p in Ω ij \, let p denote the projection o p on the plane containing. Besides, when p, we use p to denote a point on the boundary, such that p p = dist(p, ), which is the shortest distance rom p to the boundary (see Figure 1). Figure 1: illustration or the notations In short words, we deine the extension Eφ such that the values (Eφ)(p) decrease gradually when the lengths pp increase, or the lengths p p decrease. Although this property holds also or the extension designed in [8], we will use a dierent idea rom [8]. For simplicity o exposition, we make an assumption: Assumption 3.1: when p Ω ij with p, pp p p. This assumption means that each subdomain Ω k is not too thin. As we will see in Appendix, we need only to revise slightly the deinition o the extension or the general situation without this assumption. To give the exact deinition o Eφ with = Γ ij, we set Λ = {p N h (Ω ij \) : p, pp p p }. For a ace = Γ ij, deine the extension Eφ V p h (Ω ij) as ollows: 1, i p, (Eφ)(p) = 1 pp p p, i p Λ, 0, otherwise. (3.15) In particular, we have (Eφ)(p) = φ(p) when p. For rectangular ace with uniorm triangulation, the values o Eφ at some nodes are given in Figure 2. Figure 2: extension o φ or particular case 8

9 Set d = max q dist(q, ). It is clear that d is just the radium o the largest circle contained in, and d = H. It is easy to see that the extension operator E possesses the properties: (i) the support set Ω o Eφ is a simply connected domain with the size d. In act, Ω like a cone with the bottom. (ii) the calculation o Eφ possesses the optimal complexity O(n) with n = (d/h) 3 being the number o the nodes in Ω. Besides, the extension E is stable in the ollowing sense Theorem 3.1 The extension E deined by (3.15) satisies the stability condition Eφ 2 1,Ω i, Eφ 2 1,Ω j H log(h/h) = φ 2 H () ( = Γ ij ). (3.16) Since proo o the result is technical, we prove this theorem in Section 5. Remark 3.1 The deinition o the extension E not only is simple and completely explicit, but also is identical to dierent geometric shape (tetrahedron, hexahedron or other polyhedrons) o the two subdomains containing as a ace. Thus, the action o E is easy to implement. 3.2 A nearly harmonic extension o φ W 0 h(γ) Let R 0 k : W h( Ω k ) V h (Ω k ) denote the zero extension operator in the sense that R 0 k φ = φ on Ω k, and R 0 k φ vanishes at all internal nodes o Ω k or φ W h ( Ω k ). For ψ W 0 h (Γ), we have ψ = γ h, Ωk (ψ) + I 0 W k (ψ γ h, Ωk (ψ)) + Ω k γ h,(ψ γ h, Ωk (ψ))φ, on Ω k. Hereater, W k denotes the wire-basket set o Ω k, i.e., the union o all the edges o Ω k. Note that γ h,(ψ γ h, Ωk (ψ)) is just the (constant) value o ψ γ h, Ωk (ψ) at the interior nodes o. Then, we deine on each Ω k E 0 ψ = γ h, Ωk (ψ) + Rk[I 0 W 0 k (ψ γ h, Ωk (ψ))] + γ h,(ψ γ h, Ωk (ψ))eφ. (3.17) Ω k It is easy to see that E 0 ψ = ψ on Γ. The extension E 0 is nearly harmonic in the ollowing sense Proposition 3.1 The global coarse extension E 0 satisies n ω k E 0 φ 0 2 1,Ω k log(h/h) φ 0 2,Γ, φ 0 Wh 0 (Γ). (3.18) k=1 Proo. By the deinition o E 0, we have E 0 φ 0 2 1,Ω k R 0 k[i 0 W k (φ 0 γ h, Ωk (φ 0 ))] 2 1,Ω k + γ h,(φ 0 γ h, Ωk (φ 0 )) 2 Eφ 2 1,Ω k Ω k φ 0 γ h, Ωk (φ 0 ) 2 0,W k 9

10 + Using (2.11) and Friedrichs inequality, yields Ω k (H 2 φ 0 γ h, Ωk (φ 0 ) 2 0, Ω k Eφ 2 1,Ω k ). (3.19) φ 0 γ h, Ωk (φ 0 ) 2 0,W k log(h/h) φ , Ω k. (3.20) On the other hand, we deduce by (3.16) and Friedrichs inequality H 2 φ 0 γ h, Ωk (φ 0 ) 2 0, Ω k Eφ 2 1,Ω k H 1 log(h/h) φ 0 γ h, Ωk (φ 0 ) 2 0, Ω k log(h/h) φ , Ω. k Substituting (3.20) and the above inequality into (3.19), we get (3.18). 4 A substructuring method with inexact solvers This section is devoted to construction o a substructuring preconditioner with inexact solvers. The new preconditioner is based on the extension designed in the last section. 4.1 Space decomposition or V h (Ω) In this subsection, we deine a space decomposition o V h (Ω). For each subdomain Ω ij containing Γ ij, let V p h (Ω ij) be the subspace given in Subsection 2.2. Namely, V p h (Ω ij) = span{ϕ p : p N h Ω ij }. For the extension E 0 : W 0 h (Γ) V h(ω) described in the last section, deine For e E Γ, deine V 0 h (Ω) = {v h V h (Ω) : v h = E 0 φ or some φ h W 0 h (Γ)}. V h (Ωe) = span{ϕ p : p e N h }. Then, we have the space decomposition V h (Ω) = Vh 0 (Ω) + V h (Ωe) + V p h (Ω ij). e E Γ Γ ij Remark 4.1 Since the new extension E 0 described in the last section is used in Vh 0(Ω), the coarse subspace Vh 0 (Ω) is dierent rom the one in [8]. 4.2 A substructuring preconditioner Based on the space decomposition in the last subsection, we can deine a preconditioner in the standard way. Deine symmetric and positive deinite operators as ollows: the global coarse solver B 0 : Vh 0(Ω) V h 0 (Ω) satisies: the edge solver Be : V h (Ωe) V h (Ωe) satisies: (B 0 v h, v h ) = (A h v h, v h ), v h V 0 h (Ω); (4.1) (Bev h, v h ) = (A h v h, v h ), v h V h (Ωe); (4.2) 10

11 the interace solver B ij : V p h (Ω ij) V p h (Ω ij) satisies: (B ij v h, v h ) = (A h v h, v h ), v h V p h (Ω ij). (4.3) In applications, the interace solver B ij is usually chosen as a symmetric multigrid preconditioner or the restriction o A h on V p h (Ω ij). Since all the subspaces Vh 0 (Ω) and V h (Ωe) have low dimensions, the solvers B 0 and Be can be simply deined as the restriction operators o A h on their respective subspaces. Now, the desired domain decomposition preconditioner or A h is deined as B 1 h = B 1 0 Q 0 + e E Γ B 1 e Q e + Γ ij B 1 ij Q ij, (4.4) where Q 0 : V h (Ω) V 0 h (Ω), Q e : V h (Ω) V h (Ωe) and Q ij : V h (Ω) V p h (Ω ij) denote L 2 projectors. 4.3 Implementation Beore describing our algorithms, we give exact deinitions o the edge solver Be and the coarse solver B 0. the edge solver It is known that v 0,Ωk = v 0,e, v V h (Ωe) (e Ω k ). Then, (A h v, v) = ω k v 2 0,e, k Q e We can deine Be : V h (Ωe) V h (Ωe) by (Bev, w) = ( ω k ) v, w e, k Q e v V h (Ωe). w V h (Ωe). The stiness o Be is just a scaling o the mass matrix associated with e. the coarse solver As we will see in Section 5, we have As in Proposition 3.1, one can veriy that E 0 φ 2 1,Ω k = H log(h/h). v 0 h 2 1,Ω k = v 0 h γ h, Ωk (v 0 h) 2 0,W k +H log(h/h) Hence, we have or any v 0 h V 0 h (Ω) Ω k γ h,(v 0 h γ h, Ωk (v 0 h)) 2, v 0 h V 0 h (Ω). (A h vh, 0 vh) 0 = N ω k { vh 0 γ h, Ωk (vh) 0 2 0,W k + H log(h/h) γ h,(vh 0 γ h, Ωk (vh)) 0 2 }. k=1 Ω k This means that B 0 can be deined as (B 0 vh 0, w0 h ) = N ω k { vh 0 γ h, Ω k (vh 0), w0 h γ h, Ω k (wh 0) h,w k k=1 +H log(h/h) γ h,(vh 0 γ h, Ω k (vh 0)) γ h,(wh 0 γ h, Ω k (wh 0))}, Ω k vh 0 V h 0(Ω), w0 h V h 0(Ω). 11

12 Then, the stiness matrix o B 0 is calculated by B 0 = (b rl ) L0 L 0 with b rl = (B 0 (E 0 ψr), 0 E 0 ψl 0) = N ω k { ψr 0 γ h, Ωk (ψr), 0 ψl 0 γ h, Ω k (ψl 0) h,w k k=1 +H log(h/h) γ h,(ψr 0 γ h, Ωk (ψr)) 0 γ h,(ψl 0 γ h, Ω k (ψl 0))}. Ω k (r, l = 1,, L 0 ) Remark 4.2 Note that our coarse solver B 0 is dierent rom the one described in Algorithm 6.10 o [8]. The coarse solver in [8] involves N extra unknowns (see also [5]), which need to be solved by a special technique (see [32]). The coarse solver B 0 is similar with the one given in Subsection 4.3 o [7], in which stability o the coarse solver was not proved (We do not know why the article [7] cited directly the stability result o the coarse solver in [8]). The action o Bh 1 can be described by the ollowing algorithm Algorithm 4.1. For g V h (Ω), the solution u g V h (Ω) satisying can be gotten as ollows: Step 1. Computing u 0 Vh 0 (Ω) by (B h u g, v h ) = (g, v h ), v h V h (Ω) (B 0 u 0, v h ) = (g, v h ), v h V 0 h (Ω); Step 2. Computing ue V h (Ωe) in parallel by (Beue, v h ) = (g, v h ), v h V h (Ωe); Step 3. Computing u p ij V p h (Ω ij) in parallel by (B ij u p ij, v h) = (g, v h ), v h V p h (Ω ij); Step 4. Set u g = u 0 + ue + u p ij. e E Γ Γ ij Remark 4.3 The eiciency o the preconditioner B h strongly depends on the action o the coarse solver B 0. The ace basis extensions Eφ play a key role in the deinition o B 0. This is why we ocus on the design o the extension E in this paper. In the rest o this subsection, we consider an acceleration o Algorithm 4.1 (see [16] or the detailed discussion on this kind o acceleration). For convenience, set V = V h (Ω), and deine V 1 = Vh 0 (Ω) + V p h (Ω ij) and V 2 = V h (Ωe). Γ ij e E Γ Let V 2 denote the orthogonal complement o V 2 with respect to the inner product (A h, ). It is clear that V = V 1 + V 2. Let Ae : V h (Ωe) V h (Ωe) denote the restriction o A h on V h (Ωe). Deine B 1 1 = B 1 0 Q 0 + Γ ij B 1 ij Q ij. 12

13 The ollowing algorithm can be viewed as a combination between the CG method and the multiplicative Schwarz method with the above decomposition or solving (2.4). Algorithm 4.2 (multiplicative Schwarz-CG) Let u 1 V be an initial guess such that the error u h u 1 V2 (or example, u 1 can be chosen as u 1 = A 1 e Q e h V 2 ). When an e E Γ approximation u n V has been gotten, we look or u n+1 V as ollows (n 1): Step 1. Solve ε n1 V 1 by ε n1 = B1 1 ( h A h u n ). I ε n1 = 0, then the iteration is terminated; otherwise, goto Step 2: Step 2. Compute ue V h (Ωe) in parallel by (Aeue, v) = ( h A h (u n + ε n1 ), v), v V h (Ωe). Deine ε n2 V 2 by and set Step 3. Compute and ε n2 = e E Γ ue, ε n = ε n1 + ε n2. α n = ε n (ε n, A h α n 1 ) α n 1 2 A h α n 1, (α 0 = 0) u n+1 = u n + ( h A h u n, α n ) α n 2 A h α n. Remark 4.4 Step 1 in Algorithm 4.2 would be implemented by Step 1 and Step 3 in Algorithm 4.1. Since each V h (Ωe) has a very low dimension, which equals the number o nodes in the coarse edge e, Step 2 in Algorithm 4.2 is also very cheap. 4.4 Convergence The ollowing result gives an estimate o cond(bh 1 A h). Theorem 4.1 Let the extension be deined by Subsection 3.1. For the preconditioner B h deined in Subsection 4.3, we have cond(b 1 h A h) C log 2 (H/h), (4.5) where C is a constant independent o h, H and the jumps o the coeicient ω across the aces Γ ij. The convergence o Algorithm 4.2 involves the subspace V 2. Deine T = (B 1 1 A h) V 2. It is clear that T is symmetric and positive deinite with respect to the inner product (A h, ). Theorem 4.2 [16] Let the sequence u n+1 be deined by Algorithm 4.2. Then κ(t u h u n+1 Ah 2( ) 1 κ(t ) + 1 )n u h u 1 Ah, (4.6) where κ(t ) denotes the condition number o the operator T. Moreover, we have κ(t ) cond(bh 1 A h). 13

14 Remark 4.5 Algorithm 4.2 is as cheap as PCG method or solving (2.4) with the preconditioner B h. However, Theorem 4.2 indicates that the ormer has a aster convergence speed than the later. Beore proving Theorem 4.1, we need to derive a new estimate o the extension E 0. For φ W h (Γ), deine φ 0 Wh 0 (Γ) by φ 0 = γ h,(φ)i γ h,e(φ)i 0 e1 + φ(p)ϕ p. (4.7) F Γ e E Γ p V Γ Lemma 4.1 Let φ W h (Γ), and let φ 0 be deined by (4.7). Then, φ 0 γ h, Ωk (φ) 2 0,W k φ γ h, Ωk (φ) 2 0,W k, (4.8) and γ h, Ωk (φ 0 γ h, Ωk (φ)) 2 H 2 φ γ h, Ωk (φ) 2 0, Ω k. (4.9) Proo. Since we have γ h, Ωk (φ) = Then, 1 = I I 0 e1 + φ p on Ω k, Ω k e Ω k p Ω k V Γ Ω k γ h, Ωk (φ)i φ 0 γ h, Ωk (φ) = e Ω k γ h, Ωk (φ)i 0 e1 + p Ω k V Γ γ h, Ωk (φ)ϕ p on Ω k. (γ h,(φ) γ h, Ωk (φ))i1 0 + (γ h,e(φ) γ h, Ωk (φ))ie1 0 Ω k e Ω k + (φ(p) γ h, Ωk (φ))ϕ p p Ω k V Γ = γ h,(φ γ h, Ωk (φ))i γ h,e(φ γ h, Ωk (φ))ie1 0 Ω k e Ω k + (φ(p) γ h, Ωk (φ))ϕ p on Ω k. (4.10) p Ω k V Γ Furthermore, we get by the direct calculation and the inverse estimates φ 0 γ h, Ωk (φ) 2 0,W k H and φ 0 γ h, Ωk (φ) 2 0, Ω k H 2 It can be veriied that γ h,e(φ γ h, Ωk (φ)) 2 + h φ γ h, Ωk (φ) 2 0,,W k e Ω k φ γ h, Ω k (φ) 2 0,W k, (4.11) γ h,(φ γ h, Ωk (φ)) 2 + h 2 φ γ h, Ωk (φ) 2 0,, Ω k Ω k φ γ h, Ω k (φ) 2 0, Ω k. (4.12) γ h, Ωk (φ 0 γ h, Ωk (φ)) 2 H 2 φ 0 γ h, Ωk (φ) 2 0, Ω k, which, together with (4.12), gives (4.9). 14

15 Lemma 4.2 Let φ W h (Γ), and let φ 0 be deined by (4.7). Then, n ω k E 0 φ 0 2 1,Ω k log(h/h) φ 2,Γ. (4.13) k=1 Proo. It suices to estimate (3.19) more careully. It is easy to see that φ 0 γ h, Ωk (φ 0 ) = (φ 0 γ h, Ωk (φ)) γ h, Ωk (φ 0 γ h, Ωk (φ)). By the triangle inequality, we deduce and φ 0 γ h, Ωk (φ 0 ) 2 0,W k φ 0 γ h, Ωk (φ) 2 0,W k + γ h, Ωk (φ 0 γ h, Ωk (φ)) 2 0,W k φ 0 γ h, Ωk (φ) 2 0,W k + H γ h, Ωk (φ 0 γ h, Ωk (φ)) 2, φ 0 γ h, Ωk (φ 0 ) 2 0, Ω k φ 0 γ h, Ωk (φ) 2 0, Ω k + γ h, Ωk (φ 0 γ h, Ωk (φ)) 2 0, Ω k φ 0 γ h, Ωk (φ) 2 0, Ω k + H 2 γ h, Ωk (φ 0 γ h, Ωk (φ)) 2. Substituting (4.8)-(4.9) and (4.12) into the above two inequalities, yields φ 0 γ h, Ωk (φ 0 ) 2 0,W k φ γ h, Ωk (φ) 2 0,W k + H 1 φ γ h, Ωk (φ) 2 0, Ω k, (4.14) and φ 0 γ h, Ωk (φ 0 ) 2 0, Ω k φ γ h, Ωk (φ) 2 0, Ω k. (4.15) Combining (4.14) with (2.11), leads to φ 0 γ h, Ωk (φ 0 ) 2 0,W k log(h/h)h 1 φ γ h, Ωk (φ) 2 1 2, Ω k. (4.16) Plugging (4.16) and (4.15) in (3.19), and using Friedrichs inequality, leads to (4.13). Proo o Theorem 4.2. The idea o the proo is standard. But, or readers convenience, we still give a complete proo o this theorem below. One needs to establish a suitable decomposition or v h V h (Ω) v h = v 0 + ve + v ij, (4.17) e E Γ Γ ij with v 0 V 0 h (Ω), ve V h (Ωe) and v ij V p h (Ω ij). This decomposition should satisy the stability condition (B 0 v 0, v 0 ) + (Beve, ve) + (B ij v ij, v ij ) log 2 (H/h)(A h v h, v h ). (4.18) e E Γ Γ ij Set φ h = v h Γ, and deine φ 0 W 0 h (Γ) as in (4.7). For e E Γ and = Γ ij F Γ, set φe = I 0 e[φ h γ h,e(φ h )], and φ ij = I 0 [φ h γ h,(φ h )]. It is clear that φe W h (e) and φ ij W h (). It is easy to see that φ h = φ 0 + φe + φ ij. (4.19) e E Γ Γ ij 15

16 Let ve be the zero extension o φe, and set v 0 = E 0 φ 0. Then, ve V h (Ωe) and v 0 V 0 h (Ω). Let vij H V p h (Ω ij) be deined such that vij H Γ ij = φ ij, and vij H is discrete harmonic on Ω i and Ω j. Set v p k = (v h v 0 ve vij H ) Ωk. (4.20) e E Γ Γ ij Then, we have v k V p h (Ω k) by (4.19). For each k, let m k be the number o aces that belong to Ω k. Deine v ij = v H ij + v p i /m i + v p j /m j. It suices to veriy (4.18) or the unctions deined above. For convenience, set G(v h ) = (B 0 v 0, v 0 ) + (Beve, ve) + (B ij vij H, vij H ). e E Γ Γ ij Using (4.1), (4.13) and the trace Theorem, we get Moreover, we have by (4.2), (2.12) and (2.14) (B 0 v 0, v 0 ) log(h/h)(a h v h, v h ). (4.21) (Beve, ve) ve 2 0,e log2 (H/h) k Q e ω k Ω k v h 2 dp. (4.22) Here, we have used the discrete norm to derive the irst inequality or ve vanishing at all nodes outside e. Besides, we deduce by (4.3), the deinition o vij H and (2.13) (B ij vij H, vij H ) (ω i + ω j ) φ ij 2 1 H 2 00 (Γ log2 (H/h)(ω i v h 2 1,Ω i + ω j v h 2 1,Ω j ). (4.23) ij) This, together with (4.21) and (4.22), leads to G(v h ) log 2 (H/h)(A h v h, v h ). (4.24) On the other hand, it ollows by (4.20) that v p k 2 1,Ω k v h 2 1,Ω k + v 0 2 1,Ω k + ve 2 1,Ω k + vij H 2 1, Ω k. e E Γ Γ ij Combing this with (4.1)-(4.3), yields n ω k v p k 2 1,Ω k (A h v h, v h ) + G(v h ). k=1 This, together with (4.24), leads to Γ ij (B ij (v p i /m i + v p j /m j), v p i /m i + v p j /m j) n ω k v p k 2 1,Ω k log 2 (H/h)(A h v h, v h ). By the deinition o v ij, (4.23) and the above inequality, we urther get Γ ij (B ij v ij, v ij ) log 2 (H/h)(A h v h, v h ). Now, (4.18) is a direct consequence o the above inequality, together with (4.24). k=1 16

17 5 Analysis or the stability o the extension E This section is devoted to veriication o the stability condition (3.16), which has been used in Lemma 4.2. Since quasi-uniorm meshes are considered, the analysis is a bit technical. Our basic idea is to introduce a suitable approximate extension o E. This auxiliary extension is deined by positive integers, so that its stability can be veriied more easily by estimating two inite sums associated with the discrete H 1 semi-norm. 5.1 An auxiliary extension For ace = Γ ij and a node in Ω ij, let p, p, d and Λ be deined as in Subsection 4.1. Without loss o generality, we assume that h min{ pp : p N h (Ω ij \)}. For a positive number x, let [x] denote the integer part o x. Set n = [ d h ]. For any node p Ω ij \ with p, deine m p = [ pp h ] and n p = [ p p h ]. To understand the meaning o the integers m p and n p more intuitively, we image that the segments pp and p p are divided into some smaller segments with the size h. Then, m p and n p can be viewed roughly as the numbers o the division points on the segments pp and p p, respectively. By the deinition o the set Λ, it is easy to see that the positive integers m p and n p possess the properties: Property A. For each node p Λ, we have 1 m p n p n; Property B. For two integers r and k satisying 1 r k n, there are at most O(n) nodes p Λ, such that these nodes p deine the same m p = r and n p = k; Property C. For an integer k satisying 1 k n, there are at most O(m p n) nodes p Λ, such that these nodes p deine the same n p = k. For the constant-like basis unction φ = I 0 1, deine the auxiliary extension 1, i p, (E φ)(p) = 1 mp n p, i p Λ, (5.1) 0, otherwise. For the veriication o the stability (3.16), one needs to prove the ollowing two inequalities (E E )φ 2 1,Ω i d log(d/h) (5.2) and E φ 2 1,Ω i d log(d/h). (5.3) 5.2 Auxiliary results In this subsection, we estimate two inite sums, which will be used to veriy the inequality (5.3). In the rest o this section, p 1 and p 2 always denote two neighboring nodes. Since the triangulation T h is quasi-uniorm, there exists a (ixed) positive integer k 0, such that p 1 p 2 k 0 h or any two neighboring nodes p 1 and p 2. Then, it can be veriied, by the deinitions o m p and n p, that m p1 m p2, n p1 n p2 k (5.4) 17

18 Lemma 5.1 Let Λ be the set o nodes p, at which the extension E φ vanish. Namely, Λ = {p N h Ω ij : p, p Λ}. Then, h (1 m p 2 ) 2 n p2 d. (5.5) p 1 Λ p 2 Λ Note that p 1 and p 2 above denote two neighboring nodes. Proo. For a node p 2 Λ, there are at most inite nodes p 1 Λ, such that p 1 and p 2 are neighboring each other. Then, where h p 1 Λ p 2 Λ (1 m p 2 ) 2 n p2 h (1 m p 2 ) 2, (5.6) n p2 p 2 Λ Λ = {p Λ : there is p Λ such that p and p are neighboring}. For p Λ, let p Λ denote a neighboring node with p, and let p be the projection o p on the plane containing. Set Λ (1) = {p Λ : p } and Λ (2) = {p Λ : p / }. Then, It is easy to see that the set Λ (1) h p 1 Λ p 2 Λ Λ (2) Λ = Λ (1) Λ (2). contains O(n ) nodes at most. It ollows by (5.6) that (1 m p 2 ) 2 n p2 h p Λ (1) h p Λ (1) (1 m p n p ) 2 + h p Λ (2) (1 m p n p ) 2 (1 m p n p ) 2 + d. (5.7) Let p. From the deinition o Λ, we know that either p Ω ij \ or p p > p p, where p is deined as in Subsection 3.1. By Assumption 3.1, we have m p n p in any case. Then, we deduce, by (5.4), that This implies that m p m p (k 0 + 1) n p (k 0 + 1) n p 2(k 0 + 1). max{1, n p k } m p n p, p Λ (1), with k = 2(k 0 + 1). Thus, we have rom Property A and Property B p Λ (1) (1 m p ) 2 n p n k k (1 m n p ) 2 + n n p=1 m p=1 p n p=k +1 n n p n (k ) 2 + (1 m p n p=k +1 m p=n p k n p (1 m p ) 2 n m p=n p k p ) 2 n p. (5.8) 18

19 It is easy to see that n p m p=n p k (1 m p n p ) 2 n p m p=n p k (1 n p k n p ) 2 (k ) 3 Plugging this in (5.8), and note that k is a constant, leads to h p Λ (1) (1 m p ) 2 n p d (k ) 2 + (k ) 3 n n 2 n p=k +1 p Combining (5.7) with the above inequality, gives the desired result. 1. n 2 p 1 d. Lemma 5.2 The ollowing inequality holds or two neighboring nodes p 1 and p 2 Proo. For p 1 Λ, deine h p 1,p 2 Λ ( m p 1 m p 2 ) 2 n p1 n p2 d log(d/h). (5.9) Λ,p 1 = {p Λ : p is neighboring with p 1 }. Let k 0 be deined by (5.4). Then, we have rom Property A and Property B h p 1,p 2 Λ ( m p 1 m p 2 ) 2 n p1 n p2 h d h = d + d + d k 0 +1 n n p1 n p1 =1 m p1 =1 n p1 n p1 =1 m p1 =1 n n p1 =k 0 +2 m p1 =1 n n p1 ( m p1 m p 2 ) 2 n p 2 Λ p1 n p2,p 1 ( m p1 m p 2 ) 2 n p 2 Λ p1 n p2,p 1 k 0 n p1 =k 0 +2 m p1 =k 0 +1 ( m p1 m p 2 ) 2 n p 2 Λ p1 n p2,p 1 ( m p1 m p 2 ) 2 n p 2 Λ p1 n p2,p 1 = I 1 + I 2 + I 3. (5.10) It is clear that I 1 d. When m p1 k 0, we deduce rom (5.4) that m p2 2k or p 2 Λ,p 1. Note that k 0 is a constant, we get It ollows by (5.4) that n I 2 d ( 1 n n p1 =k 0 +2 p 2 Λ p 1 n 2 ) p 2,p 1 n ( ) 1 d 1 n 2 + n p1 =k 0 +2 p 1 (n p1 k 0 1) 2 d. (5.11) I 3 d n n p1 k 0 +1 k 0 +1 n p1 =k 0 +2 m p1 =k 0 +1 k= k 0 1 r= k 0 1 ( mp1 n p1 m ) 2 p 1 + r n p1 + k 19

20 = d d d n n p1 k 0 +1 k 0 +1 n p1 =k 0 +2 m p1 =k 0 +1 k= k 0 1 r= k 0 1 n n p1 n p1 =k 0 +2 m p1 =k 0 +1 n n p1 =k 0 +2 m 2 p 1 + n 2 p 1 n 2 p 1 (n p1 k 0 1) 2 n p1 (n p1 k 0 1) 2 d log n. Plugging (5.11) and the above inequality in (5.10), leads to h p 1,p 2 Λ 5.3 Proo o Theorem 4.1 It suices to veriy that ( kmp1 rn p1 n p1 (n p1 + k) ( m p 1 m p 2 ) 2 n p1 n p2 d log(d/h). Eφ 2 1,Ω i, Eφ 2 1,Ω j d log(d/h) = φ 2 H (). (5.12) Let E φ be the auxiliary extension deined by Subsection 5.1. Step 1. Veriy the inequality (5.2) Let p Λ. Then, we have Since we get Plugging this in (5.13), leads to (Eφ E φ)(p) = pp p p p [ pp h ]/[ p ]. (5.13) h pp p h /( p + 1) [ pp p h h ]/[ p h ] ( pp h + p 1)/ p h, pp p p + h [ pp p h ]/[ p h (Eφ E φ)(p) ] pp + h p p. By the inverse estimate and the discrete L 2 -norm, we get ) 2 h p p 1 n p. (5.14) This, together with (5.14), yields (E E )φ 2 1,Ω i h 2 (E E )φ 2 0,Ω i = h (Eφ E φ) 2 (p). p Λ (E E )φ 2 1,Ω i h By Property A and Property C, we get n p Λ 2 p 1. (5.15) 1 n p Λ 2 p n n n p=1 m p n 2 p n n n p=1 n p n 2 p n log n. 20

21 Plugging this in (5.15), gives (5.2). Step 2. Veriy the inequality (5.3) For simplicity o exposition, let Λ denote the set o neighboring node paring (p 1, p 2 ), which satisies (E φ )(p 1 ) (E φ )(p 2 ) 0. By the discrete H 1 semi-norm, we have E φ 2 1,Ω i = h For ease o notation, deine (p 1, p 2 ) Λ (E φ)(p 1 ) (E φ)(p 2 ) 2. (5.16) Λ, b = {p N h : p closes }. It is easy to see that the set Λ can be decomposed into several groups: (a) p 1 Λ,b and p 2 ; (b) p 1 and p 2 Λ; (c) p 1 Λ and p 2 Λ; (d) p 1, p 2 Λ. It is certain that one can also consider the inverse situation with exchanging the positions o p 1 and p 2, but this will not aect the result. It ollows by (5.16) that E φ 2 1,Ω i = h + h p 1 Λ,b p 2 p 1 Λ p 2 Λ (1 0) 2 + h p 2 Λ p 1 (1 m p 2 ) 2 + h n p2 p 1,p 2 Λ ( m p 2 ) 2 n p2 ( m p 1 m p 2 ) 2. (5.17) n p1 n p2 It is clear that the set Λ,b contains only O(n) nodes. Then, we get or two neighboring nodes p 1 and p 2 h (1 0) 2 h d h = d. (5.18) p 1 Λ,b p 2 When p 2 Λ is neighboring with some p 1, we have m p2 p 2 p 2 /h p 1 p 2 /h k 0. Besides, or any p 2 Λ, there are at most inite nodes p 1, such that p 1 and p 2 are neighboring each other. Thus, we deduce by Property A and Property B h p 1 p 2 Λ ( m p 2 ) 2 n p2 h d h k 0 n l=1 n p2 =1 Substituting (5.18)-(5.19), (5.5) and (5.9) into (5.17), yields (5.3). Step 3. Prove the desired result (5.12) Combining (5.2) and (5.3), yields In the same way, we can prove that Eφ 2 1,Ω i d log(d/h). Eφ 2 1,Ω j d log(d/h). ( l ) 2 n p2 d. (5.19) On the other hand, it can be veriied, by the discrete semi-norm in Subsection 2.2, that φ 2 H () = d log(d/h). 21

22 6 Numerical experiments In this section, we give some numerical results to conirm our theoretical results described in section 4. Consider the elliptic problem (2.1) with Ω being the cube Ω = [0, 1] 3, and the coeicient a(x, y, z) being deined by a(x, y, z) = 10 5, i x, y 0.5 or x, y 0.5; a(x, y, z) = 1, otherwise. The source unction is chosen in a suitable manner. Let Ω be decomposed into n cube subdomains with the edge length H. To illustrate wide practicality o the new method, we consider tetrahedron elements instead o hexahedron elements. Let each subdomain be divided into tetrahedron elements with the size h in the standard way, and use the usual P 1 inite element approximate space. We solve the algebraic system associated with the equation (2.4) by PCG iteration with the preconditioner B h deined in Section 4 or Algorithm 4.2. Here, each local solver B ij is chosen as the symmetric multigrid preconditioner or the restriction o A h on the subspace V 0 h (Ω ij). The iteration terminates when the relative remainder is less than 1.0D 5. The iteration counts are listed as the ollowing tables. Table 6.1 iteration counts or PCG with B h H/h H = 1/4 H = 1/6 H = 1/ Table 6.2 iteration counts or Algorithm 4.2 H/h H = 1/4 H = 1/6 H = 1/ These numerical results indicates that the convergence o the new preconditioner is stable with the subdomain number N, and depends slightly on the ratio H/h. These are just predicted by Theorem 4.1 and Theorem 4.2. Besides, the numerical results show that Algorithm 4.2 is indeed more eicient than the standard PCG. 7 Appendix In this Appendix, we revise the deinition o the extension E or the general case without Assumption

23 Let o denote the center o the largest circle 2 contained in. Then, o satisies dist(o, ) = d. Through the center o, we draw the perpendicular line L o. Let q 1, q 2 Ω ij denote the two intersection points o L with Ω ij. Set s = min{ oq 1, oq 2 }. Set r = d/s. Since Ω i and Ω j are regular and convex domains, the ratio r is uniormly bounded or every. With the same notations in Subsection 3.1, set Λ = {p N h (Ω ij \) : p, r pp p p }. For a ace = Γ ij, we revise the extension E as : 1, i p, (Eφ)(p) = 1 r pp p p, i p Λ, 0, otherwise. For such revision, the proo o Theorem 3.1 is almost the same with that in Section 5. We revise the deinition o the positive integer m p or p Ω ij \ (p ) as m p = [ r pp ], h (7.1) but reserve the deinition o the auxiliary extension E given in (5.1). We need only to revise slightly the proo o Lemma 5.1. Below is the changes needed to make. The inequality (5.4) now becomes to be m p1 m p2 rk 0 + 1, n p1 n p2 k 0 + 1, (7.2) or two neighboring nodes p 1 and p 2. Let Λ and Λ be deined in Lemma 5.1. For p Λ, let p Λ be some neighboring node with p. When p Ω ij, we deduce that r p p p p by the deinition o r and the convexity o Ω i and Ω j. Then, we have m p n p or p Λ. For p Λ, we get by (7.2) m p m p (rk 0 + 1) n p (rk 0 + 1) n p (r + 1)k 0 2. This implies that with k = (r + 1)k max{1, n p k } m p n p, p Λ, Remark 7.1 It is easy to see that the above revision is needed only or the case with r > 1, since r 1 implies that Assumption 3.1 is satisied. When r 1, the use o the original deinition o E (see Subsection 3.1) can reduce the cost or calculating Eφ. 8 Conclusions We have developed a kind o simple nearly harmonic extension or the constant-like basis unction. This extension is used to deine a coarse subspace, and then is applied to construct a substructuring preconditioner with inexact subdomain solvers. The simplicity o the extension guarantees that the resulting preconditioner is easy to implement. 2 I possesses two parallel edges, then there may exist ininite largest circles contained in. For this case, one needs to consider two such circles, which are tangent with three or more edges o. Besides, the point o can be also chosen as any central point o. Dierent choices o o will result in dierent values o the constant in the right side o (5.5). 23

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