A Non-overlapping Domain Decomposition Method with Inexact Solvers 1

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1 A Non-ovelapping Domain Decomposition Method with Inexact Solves Qiya Hu and Dudu Sun LSEC, Institute of Computational Mathematics and Scientific/Engineeing Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Abstact In this pape we ae concened with non-ovelapping domain decomposition methods fo the second-ode elliptic poblems in thee-dimensional domains. We develop a new kind of substuctuing method, in which one can use an inexact solve both in each subdomain and on each local inteface. The main ideas ae to decompose the inteface space into the sum of small local subspaces and a seies of coase subspaces spanned by constant-like basis functions, and to design a cheap appoximate hamonic extension of the constant-like function. It will be shown that the condition numbe of the peconditioned system gows only as the logaithm of the dimension of the local poblem associated with an individual substuctue, and is independent of possible jumps of the coefficient in the elliptic equation. Key wods. domain decomposition, inteface space, multilevel decomposition, constant-like function, nealy hamonic extension, peconditione, inexact solves, condition numbe AMS(MOS) subject classification 65F0, 65N30, 65N55 Intoduction Non-ovelapping domain decomposition methods (DDMs) have been shown to be poweful techniques fo solving lage-scale patial diffeential equations, especially fo solving patial diffeential equations with lage jump coefficients, and fo solving coupled patial diffeential equations. One s main task in non-ovelapping DDMs is the constuction of an efficient substuctuing peconditione fo discetization system associated with the patial diffeential equations. The constuction of this peconditione has been investigated fom vaious ways and to vaious models in liteatue, see, fo example, []-[9], []-[5], [7]-[2], [23]-[24], [26]. Most non-ovelapping DDMs studied so fa equie exact subdomain solves; we efe [25] and [29] (and the efeences cited theein). Such a equiement seveely degade the efficiency of the methods. Thee ae only a few woks studying substuctuing methods with inexact subdomain solves [3], [4] and [9]. In [3], analysis and numeical expeiments with inexact algoithms of Neumann-Diichlet type was done unde the additional assumption of high accuacy of the inexact solves. The essential difficulty is that discete hamonic extensions on each subdomain ae used in non-ovelapping domain decomposition methods. In [4], the hamonic extension on a subdomain was eplaced by a simple aveage extension, and substuctuing peconditiones with the aveage extension ae constucted. Because of such aveage extension, nealy optimal convegence can not be gotten fo these substuctuing peconditiones. To avoid hamonic extensions, [9] consideed so called appoximate hamonic basis functions, which still involve high accuacy of the inexact solves. It seems difficult to constuct a nealy optimal substuctuing peconditione with inexact solves without any additional assumption. The wok is was suppoted by Natual Science Foundation of China G07778, The Key Poject of Natual Science Foundation of China G053080, and National Basic Reseach Pogam of China G2005CB ( hqy@lsec.cc.ac.cn)

2 In the pesent pape, inspied by [9] and [4], we make a new attempt fo the design of substuctuing peconditione with inexact solves. The coe wok is the constuction of a kind of multilevel nealy hamonic basis on geneal quasi-unifom meshes. The main ingedients in ou constuction ae: (i) develop a special multilevel space decomposition to the inteface space; (ii) design a cheap nealy hamonic extension fo each basis function of the subspaces involved in the decomposition. It will be shown that the new substuctuing peconditione possesses nealy optimal convegence, which is independent of possible lage jumps of the coefficient acoss the inteface. Fo the new method, no additional assumption is equied, and the total computational complexity is optimal. The outline of the emainde of the pape is as follows. In Section 2, we intoduce some notation and ou motive. In Section 3, we pesent a multilevel decomposition fo the inteface space. The main esults on the substuctuing peconditione ae descibed in Section 4. In Section 5, we give an analysis of the convegence of the peconditione. In section 6, we pove the stability of the extension of constant-like function, which is used in Section 5. Some numeical esults ae epoted in Section 7. 2 Peliminaies 2. Domain decomposition Let Ω be a bounded polyhedon in R 3. Conside the model poblem { div(ω u) = f, in Ω, u = 0, on Ω, (2.) whee ω L (Ω) is a positive function. Let H0 (Ω) denote the standad Sobolev space, and let (, ) denote the L2 (Ω)-inne poduct. The weak fomulation of (2.) in H0 (Ω) is then given by the following. Find u H0 (Ω) such that whee (, ) is the scala poduct in L 2 (Ω), and A(u, v) = (f, v) v H 0 (Ω), (2.2) A(u, v) = Ω ω u vdp. We will apply a kind of non-ovelapping domain decomposition method to solving (2.2). Fo simplicity of exposition, we conside only the case with matching gids in this pape. Let T h = {τ i } be a egula and quasi-unifom tiangulation of Ω with τ i s being nonovelapping simplexes of size h ( (0, ]). The set of notes of T h is denoted by N h. We then define V h (Ω) to be the piecewise linea finite element subspace of H0 (Ω) associated with T h : V h (Ω) = {v H0 (Ω) : v τ P τ T h }, whee P is the space of linea polynomials. Then the finite element appoximation fo (2.2) is to find u h V h (Ω) such that A(u h, v h ) = (f, v h ), v h V h (Ω). (2.3) Let Ω be decomposed into the union of n polyhedons Ω,, Ωn, which satisfy Ω i Ω j = when i j. We assume that each Ω k can be witten as a union of boundaies of elements in T h, and all Ω k ae of size H in the usual sense (see [5] and [29]). Without loss of 2

3 geneality, we assume that the coefficient 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 tiangulation of Ω. The common pat of two neighboing subdomains Ω i and Ω j may be a vetex, an edge o a face. In paticula, we denote by Γ ij the common face of two neighboing subdomains Ω i and Ω j (i.e., Γ ij = Ω i Ω j ). The union of all Γ ij is denoted by Γ, which is called the inteface. In this pape, we choose Diichlet data as the inteface unknown. Define the opeato 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= The equation (2.3) can be witten as A h u h = f h, u h V h (Ω). (2.4) The goal of this pape is to constuct a substuctuing peconditione fo A h based on the domain decomposition descibed above. 2.2 Notations To intoduce the new method, we need some moe notations. Thoughout this pape, a subset G of Ω ae always undestood as an open set. The closue of G is denoted by Ḡ. subdomain spaces Fo subdomain Ω k, define and Set and define 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 }. inteface space and face spaces As usual, we define the (global) inteface space by W h (Γ) = {v Γ : v V h (Ω)}. Fo each Ω k, set Fo a subset G of Γ, define W h ( Ω k ) = {v Ωk : v W h (Γ)}. W h (G) = {φ h W h (Γ) : supp φ h G}. In paticula, fo G = Γ ij, the face space W h (Γ ij ) will be used epeatedly. intepolation-type opeato and constant-like basis Fo a subset G of Γ, define the intepolation-type opeato I 0 G : W h(γ) W h (Γ) as (I 0 Gφ h )(p) = { φh (p), if p N h G, 0, if p N h (Γ\G). 3

4 In paticula, we have (I 0 G)(p) = {, if p Nh G, 0, if p N h (Γ\G). If G is just the union of some elements on Γ, we call φ G = IG 0 to be constant-like basis function on G, which will be used epeatedly. integation aveage and algebaic aveage Fo a function ϕ h W h (Γ), let γ G (ϕ h ) denote the integation aveage of ϕ h on G, and let γ h,g (ϕ) denote the algebaic aveage of the values of ϕ on the nodes in G. sets of faces, edges, vetices and subdomains Fo convegence, let F Γ denote the set of all the faces Γ ij. Besides, let E Γ and V Γ denote the set of the inteio edges and the set of inteio vetices geneated by the decomposition Ω = Ωk, espectively. Fo an edge e E Γ, let Qe denote the set of the indices k of the subdomains Ω k which contain e as an edge. Namely, Define Qe = {k : e Ω k }. Ωe = k Q e Ω k, e E Γ. face inne-poducts, scaling nom and inteface nom Fo a subset G of Γ, let, G denote the L 2 inne poduct on G. In paticula, the, Γ is abbeviated as,. Let 0, G denote the nom induced fom, G. Fo a sub-faces G of Γ, let H G denote the size of G. Define the scaling nom Fo convenience, define φ 2, G = ( φ 2 2, G + H G φ 2 0, G) 2, φ H 2 (G). n φ h,γ = ( k= ω k φ h 2 2, Ω k ) 2, φh W h (Γ). discete noms and discete inne-poduct Discete noms (o semi-noms) of finite element functions will be used epeatedly in this pape, since the discete noms ae defined on a set of nodes only, and do not depend on the geometic shape of the undelying domain. We fist give definitions of thee well known discete noms (efe to [29]), which ae equivalent to thei espective continuous noms. Fo v h V h (Ω k ), the discete H seminom is defined by v h 2, h, Ω k = h 3 p i, p j N h Ω k v h (p i ) v h (p j ) 2, whee p i and p j denote two neighboing nodes. Similaly, the discete L 2 nom on an edge e of Ω k is defined by v h 2 0, h, e = h v h (p) 2. p N h e Let G Γ ij be the union of some elements. Define the discete H 2 semi-nom on W h (G) by ϕ h 2 h, 2, G = ϕ h (p) ϕ h (q) 2 p q 3, ϕ h W h (G), p N h G q N h G 4

5 whee p and q denote two diffeent nodes on G. Then, we define a discete inne-poduct. Fo each Ω k, set (φ(p) φ(q))(ψ(p) ψ(q)) φ, ψ h, Ωk = p q 3, φ, ψ W h ( Ω k ), p N h Ω k q N h Ω k whee p and q denote two diffeent nodes on Ω k. Define discete H 2 W h (Γ) by n φ, ψ h,γ = ω k φ, ψ h, Ωk, φ, ψ W h (Γ). k= inne-poduct on H 2 00 noms Let G Γ ij be the union of some elements. Fo a function ϕ h satisfying supp ϕ h G, define It is known that ϕ h 2 H 2 00 (G) = ϕ h 2 2, G + G ϕ h (x) 2 dist(x, G) ds(x). ϕ h 2 = H 200 ϕ h 2 = (G) 2, Ω i ϕ h 2 2, Ω, j whee ϕ h W h (Γ) denotes the zeo extension of ϕ h. Moeove, we have ϕ h 2 = ϕ h (x) 2 H 2 00 (G) G dis(x, G) ds(x). The coesponding discete semi-nom is defined by ϕ h 2 = ϕ h (p) 2 h, H 2 00 (G) dis(p, G). p N h G spectally equivalences Fo simplicity, we will fequently use the notations and =. Fo any two non-negative quantities x and y, x y means that x Cy fo some constant C independent of mesh size h, subdomain size d and the elated paametes. x = y means x y and y x. 2.3 Motivation We fist ecall the main ideas of the existing substuctuing peconditions. Let E k : W h ( Ω k ) V h (Ω k ) be the discete hamonic extension. Define the hamonic subspace V Γ h (Ω) = {v h V h (Ω) : v h Ωk = E k (φ h Ωk ) (k =,, n) fo some φ h V h (Γ)}. Then, we have the initial space decomposition V h (Ω) = n V p h (Ω k) + Vh Γ (Ω). k= Let A h,k : V p h (Ω k) V p h (Ω k) be the estiction of the opeato 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 symmetic and positive definite opeato which is spectally equivalent to the estiction of the opeato A h on the hamonic subspace 5

6 Vh Γ (Ω). Then, the classical substuctuing peconditione (efe to [5]) can be defined in the ough fom B h = N k= A h,k Q k + B h,γ Q Γ, (2.5) whee Q k and Q Γ denote the standad L 2 pojectos into thei espective subspaces. Fo the peconditione B h, we have (see [5]) cond(b h 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 solves A h,k. It was shown in [3] that substuctuing peconditiones with inexact solves Bh,k still possess nealy optimal convegence, if each B h,k has some spectally appoximation to A h,k (the usual spectally equivalence is not enough). Heeafte, inexact means that B h,k is only spectally equivalent to A h,k, fo example, B h,k is a multigid peconditione fo A h,k. It seems difficult to design an efficient substuctuing peconditione with completely inexact solves Bh,k, instead of A h,k itself o its appoximation. In essence, one has to modify the hamonic subspace Vh Γ(Ω) by eplacing each hamonic extension E k with anothe extension. In [4], a substuctuing peconditione ˆB h with inexact solves was been designed by eplacing each hamonic extension E k with a simple aveage extension. It has been shown that the condition numbe of the esulting peconditioned system can be estimated by cond( ˆB h A h) H/h. (2.7) In [9], anothe substuctuing peconditione with inexact solves was been designed by eplacing each hamonic extension E k with an appoximate hamonic extension. If the appoximate hamonic extension is exact enough, then cond( B h B h A h) log 2 (H/h). (2.8) The appoximate hamonic extension can be defined by appoximate hamonic basis functions. It is not pactical to compute all the appoximate hamonic basis functions. Because of this, an altenative method, which still equie high accuacy of B h,k, was consideed in [9]. The main idea of this pape is to avoid appoximate hamonic basis functions by constucting multilevel nealy hamonic basis functions on geneal quasi-unifom tiangulations T h. The new method contains thee main ingedients (efe to [2] and [3] fo the oiginal vesions): decompose each face space W h (Γ ij ) into the sum of multilevel coase subspaces and small local subspaces, such that each coase subspace is spanned by seveal constant-like basis functions as IG 0, and each local subspace contains only a few nodal basis functions; define an explicit extension fo each constant-like basis function IG 0 of coase subspaces, and use the aveage extension fo each nodal basis function of local subspaces. build a multilevel space decomposition to the global space V h (Ω) by using the decomposition of the inteface space and the basis extensions. Based on these, we constuct a multilevel substuctuing peconditione with completely inexact solves. As we will see, the new substuctuing peconditione not only possesses a nealy optimal convegence, but also possesses optimal computational complexity. 6

7 3 A multilevel space decomposition fo W h (Γ) In this section we establish a new kind of multilevel decomposition fo W h (Γ). To this end, we fist give seveal basic auxiliay esults. 3. Basic tools The following esults can be found in [29]. Lemma 3. Let e and f be an edge and a face of Ω k. Then, and φ h 0, f log 2 (H/h) φh 2, Ω k, φ h W h ( Ω k ) (3.) Ieφ 0 h 2, Ω k log 2 (H/h) φh 2, Ω, φ k h W h ( Ω k ). (3.2) The following esult can be poved as in Lemma 6.2 in [5], togethe with the standad technique. Lemma 3.2 Let e be an edge of Ω k. Then, φ h γe(φ h ) 2, Ω k log 2 (H/h) φh 2, Ω, φ k h W h ( Ω k ). (3.3) Let f denote a face Γ ij itself o a sub-face of Γ ij. We assume that f is just the union of some elements τ on the face Γ ij, and possesses the size d in the usual sense. Hee, we do not equie that f is a usual polygon. Lemma 3.3 Let f be defined above. Then, Ifφ 0 h H 2 log(d/h) φ h 00 (f),f, φ h W h (f). (3.4) 2 Poof. We can pove the esult diectly by using the discete noms descibed in Subsection 2.2 (note that f may be a nonstandad polygon). But, the poof will involve some complicated fomulas. Because of this, we ty to use anothe simple poof. Ou idea is to map f into a ectangle on the plane of f, and to use an existing esult on ectangles. Choose fou nodes p, p 2, p 3 and p 4 on f in ode. Assume that p p 2, p 2 p 3, p 3 p 4 and p 4 p has almost the same length. Let l denote the staight line though p and p 3, and let l denote the staight line though p 2 and p 4. The unique intesection of l and l is denoted by o, which is almost the baycente of f by the assumption. Make a sufficiently small ectangle D containing f, so that l and l ae just two diagonal lines of D. Moeove, we equie that o is the cente of D. It is clea that the size of D is also d. Fo each node p f (p o), daw a line l p though o and p. Let p and p denote the intesection of l p with f and with D, espectively. Define the well known pojection-type mapping F by F (p) = op op p, p N h f (p o). If o is just a node, then define F (o) = o. It is easy to see that F maps f onto D, and maps the meshes on f onto a quasi-unifom and egula meshes on D. Moeove, the fou vetices of D equal to F (p i ) (i =, 2, 3, 4), which ae just fou nodes of such esulting tiangulation Th D (with the diamete h). 7

8 Let W h ( D) denote the linea finite element space associated with T D h. Fo φ h W h ( f), define F φ h W h ( D) by By the discete nom, one can veify that (F φ h )(F (p)) = φ h (p), p N h f. I 0 fφ h It is known that (see [5] and [29]) I 0 D(F φ h ) H 2 = ID(F 0 φ h. (3.5) 00 ) H (f) 2 00 (D) H 2 00 (D) log(d/h) F φ h Plugging this in (3.5), and using the discete nom again, gives (3.4). Simila mapping with F has been used to constuct an explicit hamonic extension by [6]. 3.2 Initial space decomposition fo W h (Γ) This subsection is devoted to intoduction of an initial stable space decomposition fo W h (Γ). Thee ae vaious ways to design this kind of decomposition (efe to [5], [6] and [29]). How to design this decomposition is not the main inteest of this pape, since ou multilevel method has no essential dependence on such a concete decomposition. In the subsection we conside only an example of this kind of decomposition. Fo ease of notation, let F F Γ denote a geneic inteface Γ ij thoughout this Section. Let ϕ p denote the nodal basis function on the node p N h Ω. Define the global coase subspace W 0 h (Γ) = span{i 0 F, I 0 e, ϕ p : F F Γ, e E Γ, p V Γ }. Note that IF 0 and I0 e denotes the constant-like basis functions on the face F and the coase edge e, espectively. Let W h (e) and W h (F ) be the local spaces defined as in Subsection 2.2. Then, we have the space decomposition W h (Γ) = W 0 h (Γ) + 2,D. Wh (e) + Wh (F ). (3.6) e E Γ F F Γ The following esult gives a stability of the above space decomposition of W h (Γ). Theoem 3. Fo any φ h W h (Γ), thee exists a decomposition φ h = φ 0 + φe + φ F (3.7) e E Γ F F Γ with such that φ 0 Wh 0 (Γ); φe W h (e) and φ F W h (F ), φ 0 2,Γ + φe 2,Γ + φ F 2,Γ log 2 (H/h) φ h 2,Γ. (3.8) e E Γ F F Γ 8

9 Poof. The idea of the poof is standad. But, fo eades convenience, we still give a complete poof of this theoem below. Fo e E Γ and F F Γ, set φe = I 0 e[φ h γe(φ h )], and φ F = I 0 F [φ h γ F (φ h )]. It is clea that φe W h (e) and φ F W h (F ). Define φ 0 Wh 0 (Γ) by φ h (p), if p N h V Γ, φ 0 (p) = γe(φ h ), if p N h e (e E Γ ), γ F (φ h ), if p N h F (F F Γ ). It is easy to see that Then, we have by (3.2) φe 2,Γ = This, togethe with (3.3), yields φ h = φ 0 + k Q e φe + φ F. (3.9) e E Γ F F Γ ω k φe 2 2, Ω k log(h/h) k Q e e E Γ φe 2,Γ log2 (H/h) log2 (H/h) ω k φ h γe(φ h ) 2 2, Ω k. e E Γ k Q e n k= ω k φ h 2 2, Ω k ω k φ h 2 2, Ω k = log 2 (H/h) φ h 2,Γ. (3.0) On the othe hand, one gets by (3.4) (with f = F ) and Poincae inequality φ F 2, Ω i log(h/h) φ h γ F (φ h ) 2, Ω i log(h/h) φ h 2, Ω. (3.) i Similaly, we have φ F 2,Ω j log(h/h) φ h 2, Ω. j (3.2) Combining (3.) and (3.2), leads to Thus, φ F 2,Γ log2 (H/h)[ω i φ h 2 2, Ω i + ω j φ h 2 2, Ω j ]. F F Γ φ F 2,Γ log2 (H/h) n k= ω k φ h 2 2, Ω k = φ h 2,Γ. (3.3) It follows by (3.9) that φ 0 2,Γ φ h 2,Γ + φe 2,Γ + φ F 2,Γ e E Γ F F Γ φ h 2,Γ + φe 2,Γ + φ ij 2,Γ. e E Γ F F Γ This, togethe with (4.4) and (3.3), leads to φ 0 2,Γ log2 (H/h) φ h 2,Γ. (3.4) Now, the inequality (3.8) is a diect consequence of (4.4), (3.3) and the above inequality. In the est of this section, we constuct a multilevel space decomposition to each W h (Γ ij ). 9

10 3.3 Multilevel decomposition fo each Γ ij In this subsection, we descibe a multilevel decomposition fo each face F = Γ ij. In shot wods, we make a multilevel decomposition fo F, such that each sub-face geneated by the final level contains a few nodes. Fo a positive intege m, we fist decompose F into the union of non-ovelapping () () polygons F,, F m in the standad way. As usual, we assume that the decomposition () () is confoming, and all the polygons F have the same size d (0, H). Note that F may not be the union of some elements on F. Let J and m k (k =,, J) be given positive integes. Set m k = m m k fo k =,, J. As we will see, m k denotes the total numbe of all the sub-faces geneated fom the k-level decomposition fo F. Successively continuing the above pocedue, we get the following hieachical decompositions of F (see Figue ). the fist level decomposition F = m = F (). the second level decomposition () Let each F be futhe decomposed into the union of m 2 sub-polygons of m F () 2 = F (2) m ( )+l ( =,, m ). l= F () Thus, F = m m 2 = l= m 2 F (2) m 2 ( )+l = = F (2). Figue : A simple decomposition of F the k level decomposition fo 2 k J (k ) Afte geneating F fom the k level decomposition, we decompose each into the union of m k sub-polygons of F (k ) F (k ) m F (k ) k = F m k ( )+l ( = m k ). l= 0

11 Then, we get a multilevel decomposition of F F = m = m F () J = = = l= F (J) ( )+l = = F (J). F Sine the sub-polygons above may not be the union of some elements on F, it is inconvenient to make a detailed theoetical analysis fo the multilevel method intoduced late. Then, fo the pupose of analysis, we define a petubation F fo each polygon F as follows (see Figue 2) F = p N h F supp ϕ p, m k ; k J, whee ϕ p denotes the nodal basis function on the node p. It is clea that paticula, if F is just the union of some elements on F, then F = that these F still constitute a decomposition of F (k ) : F () F (). In. It is clea m F (k ) k = F m k ( )+l ( = m k ; k =,, J). (3.5) l= F Figue 2: the decomposition of F fo exposition Finally, we get anothe multilevel decomposition fo F F = m = m F () J = = = l= F (J) ( )+l = = F (J). Fo a fixed k, the sub-faces F ( =,, m k ) satisfy the following conditions: (a) each F is just the union of some elements on F ; (b) each F has the same size d k fo some d k (h, d k ) (set d 0 = H); (c) the union of all F ( =,, m k ) constitute a decomposition of F. Remak 3. In geneal the sub-face F is not a usual polygon yet, except fo some paticula situations. It is clea that the sub-faces F satisfying the conditions (a)-(c) can be geneated diectly if the gids on F have some paticula stuctue. Remak 3.2 In applications, we use only basis functions on the nodes in F, instead of F itself. Since F contains the same nodes with F, one needs not to actually geneate when implementing the method intoduced late (i.e., is ok.). F F

12 3.4 Multilevel decomposition fo W h (Γ ij ) In this subsection, we define a multilevel space decomposition fo each W h (Γ ij ) based on the decomposition given in the last subsection. Fo convenience, define m 0 = and F (0) = F. Fo k 0, let F denote the set of all the sub-faces geneated by the decomposition Namely, F m F k+ = F (k+) m k+ ( )+l ( = m k ). (3.6) l= = {F (k+) m k+ ( )+, F (k+) m k+ ( )+2,, F m k+ }. If the common edge of two sub-polygons geneated by the decomposition m F k+ = l= F (k+) m k+ ( )+l ( = m k ) contains a node at least, then we call the common edge to be a eal edge of the decomposition (3.6). Similaly, if the common vetex of seveal sub-polygons geneated by the decomposition m F k+ = l= F (k+) m k+ ( )+l ( = m k ) is just a node, then we call the common vetex to be a eal vetex of the decomposition (3.6). Let E and V denote the sets of eal edges and eal vetices of the decomposition (3.6). the local subspaces As befoe, let ϕ p denote the nodal basis function on the node p N h Γ. Set W h (F ) = span{ϕ p : p N h ( F \ F )} ( =,, m k ; k = 0,, J). It is clea that W h (F (0) ) = W h (F ) (m 0 = ). Moeove, fo each k J, the sum of all the local subspaces W h (F ) ( =,, m k ) gives a decomposition of Wh (F ). local coase subspaces Fo a sub-face F, define Note that Wh 0 we have W 0 h (F (F ) = span{i 0 f, I 0 e, φ p : f F ( =,, m k ; k = 0,, J )., e E, p V } ) is a subspace of Wh (F ). If both E and V ae empty set, then Wh 0 (F ) = span{if 0 : f F }. In this case, the dimensions of the subspace Wh 0 (F ) equal the numbe m k+ of the subfaces geneated by the fist decomposition of F. the final decomposition It is easy to see that m k J W h (F ) = Wh 0 (F ) + k=0 = = W h (F (J) ). (3.7) 2

13 Remak 3.3 In the above space decomposition, the spaces W h (F (J) ) ( =,, ) ae the finest local subspaces. The spaces Wh 0 (F ) ( =,, m k ), which can be viewed as extensions of Wh 0 (Γ) defined in subsection 3.2, ae the coase subspaces associated with the fist level decomposition of F vanishes at all nodes in F \ F. (k = 0,, J ). Note that each function in Wh 0 (F ) Remak 3.4 In applications, the multilevel decomposition would be geneated in a suitable manne such that both each local subspace W h (F (J) ) and each coase subspace Wh 0 (F ) has a low dimension. As will see, we usually set m k = m (a fixed positive intege). 3.5 Stability of the multilevel space decomposition Fo convenience, set W 00 h (Γ) = W 0 h (Γ) + e E Γ W h (e). Fom (3.6) and (3.7), we get a multilevel space decomposition of W h (Γ) m k W h (Γ) = Wh 00 (Γ) + J [ Wh 0 (F ) + F F Γ k=0 = = W h (F (J) )]. This subsection is devoted to analysis of the stability of the above multilevel space decomposition. Befoe giving the main esult, we pove two auxiliay esults fo a geneic face F F Γ. Fo simplicity of exposition, we assume that both E and V ae empty set fo each F, wee defined in the last subsection. Then, we need only to conside whee E and V sub-face intepolations in the constuction of a stable multilevel decomposition. If the assumption does not hold, we need also to conside edge intepolations as in Subsection 3.2, and to make some obvious modifications in the following constuction. But, this change will not incease any essential difficulty of the analysis. We fist constuct a decomposition fo the geneic face F. Fo k satisfying k J, let I 0 F Subsection 2.2 with G = F denote the intepolation-type opeato defined in ( =,, m k ). Fo ease of notation, set θ = I 0 F ( =,, m k ). Unde the assumption mentioned above, the union of F m k ( )+,, F m k ( )+m k gives an open cove of F (k ). Then, the opeatos satisfy m k l= θ m k ( )+l : W h(γ) W h (F m k ( )+l ) W h (F (k ) ) (l =,, m k ) θ m k ( )+l = i on W h (F (k ) ) ( =,, m k ; k =,, J). (3.8) Fo φ h W h (Γ), let φ F W h (F ) be defined in the poof of Theoem 3.. Fo convenience, set φ (0) = φ F. Define φ = θ [φ h γ F (φ h )] W h (F ) ( k J; =,, m k ) (3.9) 3

14 and φ (k ), 0 = φ (k ) m k l= φ m k ( )+l ( k J; =,, m k ). (3.20) By the definition of φ F, the equality (3.9) is also valid fo k = 0 ( = ). The following esult indicates that they constitute a decomposition of φ F. Lemma 3.4 Let φ (J) and φ (k ), 0 (k =,, J) be defined above. Then, Moeove, we have φ (k ), 0 W 0 h (F (k ) ) ( k J; =,, m k ). (3.2) φ F = Poof. By (3.9) and (3.8), we have φ (k ) J = θ (k ) = m k = m k k=0 = φ, 0 + = [φ h γ (k ) F (φ h )] θ m k ( )+l θ(k ) l= m k θ l= φ (J). (3.22) [φ h γ (k ) F (φ h )] m k ( )+l [φ h γ (k ) F (φ h )]. Substituting this into (3.20), and using (3.9) fo φ m k ( )+l, yields φ (k ), 0 = = m k θ l= m k l= m k ( )+l [φ h γ (k ) F (φ h )] θ m k ( )+l [γ F m k ( )+l m k l= (φ h ) γ (k ) F (φ h )]. θ m k ( )+l [φ h γ F (φ h )] m k ( )+l Since γ F (φ h ) γ (k ) m k ( )+l F (φ h ) is a constant, we futhe get φ (k ) m k, 0 = [γ F l= m k ( )+l This gives (3.2). Using (3.20) fo k = and k = 2, we deduce φ F = φ (0) = φ (0) m, 0 + It follows by (3.5) that = m = l= (φ h ) γ (k ) F (φ h )]θ m k ( )+l. φ () = φ (0) m, 0 + m 2 = m 2 φ (2) m 2 ( )+l = φ () m, 0 + = φ (2). m 2 = l= Substituting this into (3.23), and using (3.20) epeatedly, yields φ F = φ (0) m, 0 + = m 2 φ (), 0 + = m 3 φ (2), 0 + = φ (3) = φ (2) m 2 ( )+l. (3.23) 4

15 = φ (0) m, 0 + φ () =, = φ (J ), 0 + = φ (J). This implies (3.22). The following esult shows the stability of the decomposition (3.22). Lemma 3.5 Let φ, 0 and φ(j) be defined in (3.9) and (3.20). Then, J m k k=0 = φ, 0 2 H 200 (F ) + = Poof. It follows by (3.20) that φ (J) 2 H 200 J[ + log(h/h)]2 φ h 2 (F ) 2, F (J ). (3.24) φ (k ), 0 2 H 2 (k ) 00 (F ) 2( φ (k ) 2 m k + H 2 (k ) 00 (F ) l= φ(k ) 2 m k + H 200 (k ) (F ) l= φ m k ( )+l 2 ) H 200 (k ) (F ) φ m k ( )+l 2 H 200 (F m k ( )+l ). (3.25) One needs to estimate each semi-nom in the ight side of the above inequality. Using (3.4) with f = F (k ), yields (d 0 = H) Then, φ (k ) 2 H 2 (k ) 00 (F ) m k = φ (k ) 2 H 200 (k ) (F ) log2 (d k /h) φ h γ (k ) F (φ h ) 2 2, F, =,, m k. log2 (d k /h) φ h 2 2, F log(d k /h) φ h 2 ( k J). (3.26), F Let =,, m k ; l =,, m k. By (3.4) with f = F m k ( )+l, one can veify that φ m k ( )+l 2 H 2 00 (F 2 m k ( )+l log 2 (d k /h) φ h γ F m k ( )+l ) log2 (d k /h) φ h 2. 2, F m k ( )+l (φ h ) 2 2, F m k ( )+l This, togethe with (3.5) and the discete H 2 semi-nom, leads to m k φ m k ( )+l 2 l= H 2 00 (F m k ( )+l ) log 2 (d k /h) φ h 2 2, F (k ). We futhe get m k = m k φ m k ( )+l 2 l= H 2 00 (F Combining (3.25) with (3.26)-(3.27), leads to log 2 (d k /h) φ h 2 ( k J). (3.27) m k ( )+l ), F 2 m k = φ (k ), 0 2 H 2 (k ) 00 (F ) m k = φ (k ) 2 H 2 (k ) 00 (F ) 5 + m k = m k l= φ m k ( )+l 2 H 200 (F m k ( )+l )

16 Thus, log2 (d k /h) φ h 2 (k =,, J)., F 2 Then, J m k φ, 0 2 k=0 = H 2 00 (F ) J k=0 log 2 (d k /h) φ h 2 2, F J log2 (H/h) φ h 2 2, F. (3.28) On the othe hand, using (3.9) (with k = J) and (3.4), we deduce φ (J) 2 H 2 (J) 00 (F ) φ (J) 2 = H 2 (J) 00 (F ) log2 (d J /h) φ h 2 2, F (J) log2 (d J /h) φ h 2 2, F. Now, the inequality (3.24) follows by (3.28) and and the above inequality. Now, we give the stability of the multilevel space decomposition of W h (Γ). Fo φ h W h (Γ), let φ 0 and φe be defined by Subsection 3.2. Fo simplicity of exposition, we define φ 00 = φ 0 + e E Γ φe. The decomposition (3.7) can be witten as φ h = φ F F Γ φ F. (3.29) Then, the following esult is a diect consequence of Lemma 3.4 and Lemma 3.5. Theoem 3.2 Fo any φ h W h (Γ), let φ 00 Wh 00 (Γ) be defined above. Then, thee exist functions φ ij,, 0 W 0 h (Γ ij, ) (0 k J ; =,, m k) and φ (J) ij, W h (Γ (J) ij, ) ( =,, ), such that Moeove, we have m k φ h = φ 00 + Γ ij J m k [ k=0 = φ ij,,0 + = φ (J) ij, ]. (3.30) J [ φ ij,,0 2,Γ + φ (J) ij, 2,Γ] J[ + log(h/h)] 2 φ h 2,Γ (J ). (3.3) Γ ij k=0 = = Remak 3.5 We conjectue that the facto J in (3.3) (and (3.24)) can be dopped. Unfotunately, we fail to pove this conjectue. 4 A substuctuing method with inexact subdomain solves This section is devoted to constuction of a new substuctuing peconditione with inexact subdomain solves. The new peconditione is based on a multilevel decomposition of V h (Ω). This multilevel decomposition of V h (Ω) depends on the multilevel decomposition of W h (Γ) developed in the last section and vaious nealy hamonic extension opeatos. 6

17 4. Nealy hamonic extensions In this subsection, we define vaious extension opeatos, which will play a key ole in the new peconditione. As in the last section, let F denote a face Γ ij, and let F be defined in Subsection 3.3. Fo convenience, we use f to denote the face F o a sub-face F ( =,, m k ; k = 0,, J). Thoughout this section, we always use φf = If 0 W h (F ) to denote the constant-like basis function on f. an explicit extension fo the constant-like basis function φf Since f may not be a polygon, a stable extension of the constant-like basis φf is tick to design. Let of be a cental node on f, and let df denote the size of f in the standad sense (df = d k if f = F ). Fo a node p in Ω ij, let p denote the pojection of p on f. Besides, we use l p to denote the staight line dawing fom of to p (if p of), and use q f to denote the intesection point of l p with f (see Figue 3). Figue 3: illustation fo the notations In shot wods, we define the extension Efφf such that the values (Efφf)(p) decease gadually when the length pp o ofp inceases. To give the exact definition of Efφf, we set Λ () f = {p N h Ω ij : p = of, pp df} and Λ (2) f = {p N h Ω ij : p f, p of, pp p df q ofq }. Fo a face o a sub-face f, define the extension opeato Ef as φf(p), if p f, pp d (Efφf)(p) = f, if p Λ() f, o f q d f pp p q, if p Λ(2) f, 0, othewise. (4.) Fo ectangula face f with unifom tiangulation, the values of Efφf at some nodes ae given in Figue 4. 7

18 Figue 4: extension of φf fo paticula case It is easy to see that the extension opeato Ef possesses the popeties: (i) the suppot set Ωf of Efφf is a simply connected domain with the size df; (ii) the implementation of Ef possesses the optimal computational complexity O(nf) with nf = (df/h) 3 being the numbe of the nodes in Ωf. Besides, the extension Ef is nealy hamonic in the following sense Theoem 4. The extension Ef defined by (4.) satisfies the stability condition Efφf 2,Ω i, Efφf 2,Ω j φf 2 H 2 00 (f). (4.2) The poof of this theoem will be given in Section 6. extensions associated with the finest sub-faces Fo F = Γ ij, define E (J) ij, : Wh (F (J) ) V h (Ω ij ) as follows (f = F (J) (E (J) ij, ϕ)(p) = ϕ(p), if p f, γ h,f(ϕ), if p Λ () f Λ(2) f, 0, othewise. ) (p N h Ω ij, ϕ W h (F (J) )) (4.3) an extension on the global coase space Let Rk 0 : W h( Ω k ) V h (Ω k ) denote the zeo extension opeato in the sense that Rk 0φ = φ on Ω k, and Rk 0φ vanishes at all intenal nodes of Ω k fo φ W h ( Ω k ). Fo ψ Wh 0 (Γ), we have ψ = γ h, Ωk (ψ) + I 0 W k (ψ γ h, Ωk (ψ)) + f Ω k γ h,f(ψ γ h, Ωk (ψ))φf, on Ω k. Heeafte, W k denotes the wie-basket set of Ω k, i.e., the union of all the edges of Ω k. Then, we define on each Ω k E 0 ψ = γ h, Ωk (ψ) + Rk[I 0 W 0 k (ψ γ h, Ωk (ψ))] + γ h,f(ψ γ h, Ωk (ψ))efφf. (4.4) f Ω k It is easy to see that E 0 ψ = ψ on Γ. extensions on the local coase spaces Let G denote the union of the eal edges and the eal vetices geneated by the fist decomposition of F (see Subsection 3.4). Fo ψ Wh 0 (F ), let af(ψ) denote the (constant) value of ψ at the inteio nodes in f. It is easy to see that ψ = I 0 G ψ + f F af(ψ)φf, on F. 8

19 Note that I 0 ψ 0 if G G is an empty set. Then, we define on Ω l (l = i, j) E ij,, 0 ψ = R0 l (I 0 ψ) + af(ψ)efφf. (4.5) G f F As we will see in Section 5, all the extension opeatos defined above keep the stability of the exact hamonic extension, and so they ae called nealy hamonic extensions. 4.2 Multilevel decomposition fo V h (Ω) In this subsection, we define a seies of nealy hamonic subspaces of V h (Ω) by using the extension opeatos descibed in the last subsection. Fo e E Γ, let Ee : Wh (e) V h (Ω) denote the standad zeo extension opeato. Define V h (Ωe) = {v h V h (Ω) : v h = Eeφ h fo some φ h W h (e)}. Namely, V h (Ωe) = span{ϕ p : p N h e}. Fo the extension E 0 : W 0 h (Γ) V h(ω), define V 0 h (Ω) = {v h V h (Ω) : v h = E 0 φ fo some φ h W 0 h (Γ)}. As befoe, let F = Γ ij. Fo each F (k = 0,, J; =,, m k ), let E ij,, 0 : Wh 0 (F ) V h (Ω ij ) (k =,, J ) and E (J) : Wh (F (J) ) V h (Ω ij ) be the extension opeatos defined in the last subsection. Set Ω ij, = ψ Wh 0 (F ) supp E ij,, 0ψ (k = 0,, J ) and Define the subspaces Ω (J) ij, = ψ W h (F (J) ) supp E (J) ψ. and V 0 h (Ω ij, ) = {v h V h (Ω ij ) : v h = E ij,, 0 φ h fo φ h W 0 h (F )} (k =,, J ), Ṽ h (Ω (J) ij, ) = {v h V h (Ω ij ) : v h = E (J) ij, φ h fo φ h W h (F (J) )}. The functions in the subspaces defined in this subsection ae not hamonic yet, but they still keep the enegy stability (see Section 5). Thus, the basis functions of these subspaces ae called nealy hamonic basis functions. Since we have W h (Γ) = Wh 0 (Γ) + J Wh (e) + [ Wh 0 (F ) + e E Γ F F Γ k=0 = = the space decomposition holds V h (Ω) = n V p h (Ω k) + Vh 0 (Ω) + V h (Ωe) + e E Γ k= m k J m k [ Γ ij k=0 = W h (F (J) )], Vh 0 (Ω ij, ) + = Ṽ h (Ω (J) ij, )]. 9

20 4.3 New substuctuing peconditione Based on the space decomposition in the last subsection, we can define a peconditione in the standad way. Define symmetic and positive definite opeatos as follows: the subdomain solve B k : V p h (Ω k) V p h (Ω k) satisfies: (B k v h, v h ) Ωk = ω k Ω k v h 2 dp, v h V p h (Ω k); (4.6) the global coase solve B 0 : Vh 0(Ω) V h 0 (Ω) satisfies: the edge solve Be : V h (Ωe) V h (Ωe) satisfies: (B 0 v h, v h ) = (A h v h, v h ), v h V 0 h (Ω); (4.7) (Bev h, v h ) Ω e = (A hv h, v h ), v h V h (Ωe); (4.8) local coase solve B ij,, 0 : V h 0(Ω ij, ) V h 0(Ω ij, ) (k = 0,, J ) satisfies (B ij,, 0 v h, v h ) = (A h v h, v h ), v h Vh 0 (Ω ij, ); (4.9) the finest local solve (J) (J) (J) B ij, : Ṽh(Ω ij, ) Ṽh(Ω ij, ) satisfies (B (J) ij, v h, v h ) = (A h v h, v h ), v h (J) Ṽh(Ω ij, ). (4.0) In applications, the subdomain solve B k is usually chosen as a symmetic multigid peconditione. Since all the subspaces Vh 0(Ω), V h(ωe), Vh 0(Ω (J) ij, ) and Ṽh(Ω ij, ) have low dimensions, the solves B 0, Be, B (J) ij,, 0 and B ij, can be simply defined as the estiction opeatos of A h on thei espective subspaces. Now, the desied multilevel peconditione fo A h is defined as B J = + Γ ij n Bk Q k + B0 Q 0 + Be Q e k= e E Γ J m k [ (B ij,, 0 ) Q ij,, 0 + k=0 = = ( B (J) ij, ) Q (J) ij, ], whee Q k : V h (Ω) V p h (Ω k), Q 0 : V h (Ω) V 0 h (Ω), Q e : V h (Ω) V h (Ωe), Q ij,, 0 : V h (Ω) Vh 0(Ω ij, ) (k = 0,, J ), and Q(J) ij, : V (J) h(ω) Ṽh(Ω ij, ) denote L2 pojectos. 4.4 Implementation Let B k, B 0, Be, B ij,, 0 (k = 0,, J ) and B(J) ij, be defined as in the last subsection. The action of BJ can be descibed by the following algoithm Algoithm 4.. Fo g V h (Ω), the solution u g V h (Ω) satisfying (B J u g, v h ) = (g, v h ), v h V h (Ω) can be gotten as follows: Step. Computing u p k V p k (Ω k) in paallel by (B k u p k, v h) Ωk = (g, v h ) Ωk, v h V p k (Ω k); 20

21 Step 2. Computing u 0 Vh 0 (Ω) by (B 0 u 0, v h ) = (g, v h ), v h V 0 h (Ω); Step 3. Computing ue V h (Ωe) in paallel by (Beue, v h ) = (g, v h ), v h V h (Ωe); Step 4. Computing u (l) ij,, 0 V h 0(Ω(l) ij, ) (l = 0,, J ) in paallel by (B (l) ij,, 0 u(l) ij,, 0, v) = (g, v h) (l) Ω, v h Vh 0 (Ω (l) ij, ); ij,, 0 Step 5. Computing ũ (J) (J) ij, Ṽh(Γ ij, ) in paallel by Step 6. Set (B (J) ij, ũ(j) ij,, v (J) h) = (g, v h ) (J) Ω, v h Ṽh(Ω ij, ); ij, u g = n u p k + u 0 + ue + e E Γ k= J m k [ Γ ij l= = u (l) ij,, 0 + = ũ (J) ij, ]. Remak 4. Befoe implementing the above algoithm, one needs to compute the nealy hamonic basis E 0 φ h, Eeφ h, E ij,, 0 φ h and E (J) ij, φ h fo each basis function φ h of Wh 0(Γ), W h (e), Wh 0 (F, 0 ) o of Wh (F (J) ) (F = Γ ij ). As we will see in the next subsection, the complexity of computing all the extension is optimal. When all the hamonic basis functions ae gotten, the cost fo implementing Step 2-Step 5 is vey cheap, since each subspace involved in these steps has vey low dimension. Algoithm 4. can be also descibed in tem of inteface solves. To this end, we need to define a seies of solves on inteface subspaces. The global coase solve A symmetic opeato M 0 : Wh 0(Γ) W h 0 (Γ) satisfying M 0 v h, v h = v h 2, Γ, v h W 0 h (Ω). The edge solve A symmetic opeato Me : W h (e) : W h (e) satisfying Mev h, v h e = v h 2,Γ, v h W h (e). The local coase solve A symmetic opeato M ij,, 0 : W 0 h The finest local solve A symmetic opeato M (J) (F ) Wh 0 (F ) (F = Γ ij ) satisfying M ij,, 0 v h, v h = F (ω i + ω j ) v h 2 2,Γ, v h W 0 ij h (F ). ij, : Wh (F (J) ) W h (F (J) ) (F = Γ ij ) satisfying M (J) ij, v h, v h = (J) F (ω i + ω j ) v h 2 2,Γ, v h W ij h (F (J) ). 2

22 All the above opeatos can be defined by the discete H 2 inne-poduct (see Subsection 2.2) on thei espective subspaces of W h (Γ) (see [4] and [3] fo the details). Unde suitable assumptions, the local opeatos can be also defined explicitly (see [2]). The following algoithm can be viewed as a vaiant of Algoithm 4. Algoithm 4.2. Fo g V h (Ω), the solution u g V h (Ω) satisfying (B J u g, v h ) = (g, v h ), v h V h (Ω) can be gotten as follows: Step. Computing u p k V p k (Ω k) in paallel by Step 2. Computing φ 0 Wh 0 (Γ) by (B k u p k, v h) Ωk = (g, v h ) Ωk, v h V p k (Ω k); M 0 φ 0, ψ h Γ = (g, v h ), v h = E 0 ψ h, ψ h W 0 h (Γ); Step 3. Computing φe W h (e) in paallel by Meφe, ψ h e = (g, v h ) Ω e, v h = Eeψ h, ψ h W h (e); Step 4. Computing φ (l) ij,, 0 W h 0 (l) (F ) (l = 0,, J ) in paallel by (F = Γ ij ) M (l) ij,, 0 φ(l) ij,, 0, ψ h (l) F = (g, v h ) (l) Ω, v h = E (l) ij,, 0 ψ h, ij,, 0 ψ h W 0 h (F (l) ); Step 5. Computing φ (J) ij, W h (F (J) ) in paallel by (F = Γ ij ) M (J) (J) ij, φ ij,, ψ h (J) F = (g, v h ) (J) Ω, v h = E (J) ij, ψ h, ij, ψ h W h (F (J) ); Step 6. Computing u 0 = E 0 φ 0, ue = Eeφe, u (l) ij,, 0 = E(l) ij,, 0 φ(l) ij,, 0 (l = 0,, J ) and Step 7. Set u g = k= ũ (J) ij, = E(J) ij, φ(j) ij,. n u p k + v 0 + ue + e E Γ J m k [ Γ ij l= = u (l) ij,, 0 + = ũ (J) ij, ]. 4.5 Computational complexity When implementing an existing substuctuing method with exact subdomain solves, one needs to compute hamonic extension E k ϕ h V h (Ω k ) fo some ϕ h W h ( Ω k ) (k =,, n). Since each Ω k has O((H/h) 3 ) nodes, the optimal complexity fo computing all such extensions is O(n (H/h) 3 ) (but, thee seems no optimal hamonic extension in liteatue, except that the tiangulation T h possesses paticula stuctue). Fo ou substuctuing method with inexact subdomain solves, we need not to compute such hamonic extensions, but we have to compute all the nealy hamonic basis functions befoe implementing Algoithm 4. (o Algoithm 4.2). A natual question is: whethe the complexity fo computing all the nealy hamonic basis functions is much geate than the complexity 22

23 fo computing the oiginal hamonic extensions? If yes, the advantage of inexact subdomain solves is not obvious. The following esult gives an answe to this question. Let N(J) denote the total complexity fo computing all the nealy hamonic basis functions. Poposition 4.. Let m 4 be a given positive intege. If J and m k ae defined by then J = log m (H/h) and m = = = m, (4.) N(J) n (H/h) 3, (4.2) which is optimal. Moeove, the complexity fo implementing Step 3-Step 5 in Algoithm 4. (o Step 3 -Step 5 in Algoithm 4.2) equals O((H/h) 2 ) fo each Γ ij, which is also the optimal. Poof. We fist deive a geneal estimate of N(J). It is clea that the global coase space Wh 0 (Γ) has the dimensions O(n). Using (4.4), togethe with the popety (ii) in Subsection 4., we know that the cost fo computing E 0 φ h with all basis functions φ h of Wh 0 (Γ) equals N 0 = O(n (H/h) 3 ). (4.3) Since Ee is chosen as the simple zeo extension, the cost fo computing Eeφ h with all basis functions φ h of each W h (e) is Ne = O(H/h). Note that the numbe of the coase edges e E Γ equals to O(n), we have e E Γ Ne = O(n H/h). (4.4) Fo a face F = Γ ij, let n ij denote the numbe of the nodes in Γ ij. It is easy to see that the numbe of nodes in F is about n ij /m k, and so the numbe of nodes in Ω ij, is about (n ij /m k ) 3 2 with m 0 = (k = 0,, J). By the definition of Wh 0 ), the dimensions of each Wh 0 (F ) equal O(m k+ ) (k = 0,, J ). Moeove, the numbe of the local coase subspaces Wh 0 (F ) equals m k (k = 0,, J ). Besides, the dimensions of each W h (F (J) ) equal n ij /. Since the cost fo computing each local hamonic extension is optimal, the (J) ij, ) and Ṽh(Ω ij, ) is (F complexity fo computing all the hamonic basis functions of Vh 0(Ω about J k=0 This, togethe with (4.3) and (4.4), leads to N(J) n ( H h )3 + n H J h + n { m k [( n ij m k ) 3 2 mk+ ] + [( n ij ) 3 2 n ij ]. (4.5) k=0 m k [( n ij m k ) 3 2 mk+ ] + [( n ij ) 3 2 n ij ]}. (4.6) Now, we veify the desied esult by (4.6). By the choices of J and m k in (4.), we have m k = m k, m k+ = m (k = 0,, J ) and = = (H/h) 2. Then, m k [( n ij m k ) 3 2 mk+ ] = m n 3 2 ij ( m ) k and [( n ij ) 3 2 n ij ] = n 5 2 ij ( H h ) 3. 23

24 Substituting these into (4.6), and note that n 2 ij = O(H/h), we deduce (4.2). The numbe of subspaces Vh 0(Ω (J) ij, ) and Ṽh(Ω ij, ) equals m + m 2 + = + = O((H/h) 2 ) = n ij. Note that each subspace has almost the fixed dimension m, we get the second conclusion. 4.6 Convegence Now, we give an estimate of cond(bj A h). Theoem 4.2 Let J and m k be defined as in Poposition 4., and let the extensions be defined by Subsection 4.. Fo the peconditione B J defined in Subsection 4.3, we have cond(b J A h) CJ log 3 (H/h), (4.7) whee C is a constant independent of h, H, d k and the jumps of the coefficient ω acoss the faces Γ ij. A poof of the above theoem will be given in the next section. 5 Analysis This section is devoted to a poof of Theoem 4.. We fist to pove seveal auxiliay esults, which involve stabilities of the extensions defined in Subsection Auxiliay esults The following esult can be veified diectly by the discete semi-noms descibed in Subsection 2.2. Lemma 5. The coase edge extension Ee satisfies ω k Eeφe 2,Ω k φe 2,Γ, φe W h (e). (5.) k Q e Lemma 5.2 Let m k and J be defined in Poposition 4.. Fo F = Γ ij, we have ω i E (J) ij, ϕ h 2,Ω i + ω j E (J) ij, ϕ h 2,Ω j φ h 2,Γ, ϕ h W h (F (J) ). (5.2) Poof. Using the discete H semi-nom of finite element functions, togethe with the definition of the extension E (J) ij,, yields E (J) ij, ϕ h 2,Ω i h[ (ϕ h (p) γ h,f(ϕ h )) 2 + ( d J h )2 γ h,f(ϕ h ) 2 ] p f N h h [ ϕ h γ h,f(ϕ h ) 2 0,f + ϕ h 2 0,f]. (5.3) Since ϕ h vanishes on Γ ij \f, we get by Fiedich s inequality ϕ h γ h,f(ϕ h ) 2 0,f, ϕ h 2 0,f d J ϕ h 2 2, Ω i. (5.4) 24

25 It is easy to see that d J h = (n ij ) 2 = H h /(m ) 2. Then, we deduce by the assumption d J /h = O(). Plugging (5.4) in (5.3), leads to Similaly, we have E (J) ij, ϕ h 2,Ω i ϕ h 2 2, Ω i. E (J) ij, ϕ h 2,Ω j ϕ h 2 2, Ω j. Then, we deduce (5.2) by the definition of,γ. Lemma 5.3 The global coase extension E 0 satisfies n ω k E 0 φ 0 2,Ω k log(h/h) φ 0 2,Γ, φ 0 Wh 0 (Γ). (5.5) k= Poof. By the definition of E 0, we have E 0 φ 0 2,Ω k R 0 k[i 0 W k (φ 0 γ h, Ωk (φ 0 ))] 2,Ω k + f Ω k γ h,f(φ 0 γ h, Ωk (φ 0 )) 2 Efφf 2,Ω k. (5.6) Using the discete nom, togethe with (3.) and Poincae inequality, yields On the othe hand, we have Rk[I 0 W 0 k (φ 0 γ h, Ωk (φ 0 ))] 2,Ω k φ 0 γ h, Ωk (φ 0 ) 2 0,W k log(h/h) φ 0 2 2, Ω. (5.7) k γ h,f(φ 0 γ h, Ωk (φ 0 )) 2 H 3 φ 0 γ h, Ωk (φ 0 ) 2 0, Ω k and (by (4.2)) Thus, Efφf 2,Ω k φf 2 H 2 00 (f) = H log(h/h). γ h,f(φ 0 γ h, Ωk (φ 0 )) 2 Efφf 2,Ω k H 2 log(h/h) φ 0 γ h, Ωk (φ 0 ) 2 0, Ω k log(h/h) φ 0 2 2, Ω. k Substituting (5.7) and the above inequality into (5.6), we get (5.5). Lemma 5.4 Let F be associated with F = Γ ij. The local coase extensions E ij,, 0 satisfy ω i E ij,, 0 φ h 2,Ω i + ω j E ij,, 0 φ h 2,Ω j φ h 2,Γ, φ h W 0 h (F ) (k =,, J ). (5.8) 25

26 Poof. Without loss of geneality, we assume that F decomposed into fou quadilateals in the standad way. It follows by (4.5) that E ij,, 0 φ h 2,Ω i Rl 0 (I 0 φ G h ) 2,Ω i + f F is a quadilateal, which is just af(φ h ) 2 Efφf 2,Ω i. (5.9) We fist estimate Efφf 2,Ω i fo each f F. Using (4.2) and the discete nom given in Subsection 2.2, yields Set (see Figue 5) Efφf 2,Ω i φf 2 = H 2 φf 2 = 00 (f) h, H 2 00 (f) p f N h dis(p, f). (5.0) Lf = f F, D = {p f : dis(p, f) = dis(p, Lf)} and D = f\d. Then, dis(p, f) = p f N h Figue 5: illustation to Lf, D and D dis(p, Lf) + p D N h dis(p, f). (5.) p D N h Since the tiangulation is quasi-unifom, the domains D and D have the same size. Fo each p D N h, thee exist a p D N h such that Thus, we have by (5.) dis(p, f) p f N h Plugging this in (5.0), leads to af(φ h ) 2 Efφf 2,Ω i dis(p, f) > dis(p, f) = dis(p, Lf). dis(p, Lf) = p D N h p D N h af(φ h ) 2 dis(p, F 26 p D N h dis(p, F ). ) p F N h φ 2 h (p) dis(p, F ).

27 We futhe get by the discete nom af(φ h ) 2 Efφf 2,Ω i F φ 2 h (p) dis(p, F ) ds(p) φ h 2. (5.2) H 2 00 (F ) Now, we estimate Rl 0(I0 φ G h ) 2,Ω i. It can be veified, by the discete nom again, that R 0 l (I 0 G This, togethe with the elation yields R 0 l (I 0 G I 0 G φ h ) 2,Ω i φ h 2 H Then, we obtain by the above poof R 0 l (I 0 G φ h ) 2,Ω i I 0 φ G h 2. H 2 00 (F ) φ h = φ h 2 00 (F ) Plugging this and (5.2) in (5.9), leads to Similaly, we have E f F + f F φ h ) 2,Ω i φ h 2 H ij,, 0 φ h 2,Ω i φ h 2 H E ij,, 0 φ h 2,Ω j φ h 2 H Combining the two inequalities, gives (5.8). 5.2 Poof of Theoem 4.2 af(φ h )φf, af(φ h ) 2 φf 2. H 2 00 (F ) 2 00 (F ) 2 00 (F ) (F ) One needs to establish a suitable decomposition fo v h V h (Ω) with and v h = n v k + v 0 + ve + e E Γ k= J m l [ Γ ij l=0 =.. v (l) ij,,0 + = v k V p h (Ω k) (k =,, n), v 0 V 0 h (Ω), ve V h (Ωe), v (l) ij,, 0 V h 0 (Ω (l) (J) ij, ) (l = 0,, J ) and v(j) ij, Ṽh(Ω ij,. This decomposition should satisfy the stability condition n (B k v k, v k ) Ωk + (B 0 v 0, v 0 ) + k= + Γ ij J m l (B (l) ij,, 0 v(l) ij,, 0, v(l) l=0 = [ (Beve, ve) Ω e e E Γ ij,, 0 ) + (B (J) = ṽ (J) ij, ], (5.3) ij, ṽ(j) ij,, ṽ(j) ij, )] 27

28 J log3 (H/h)(A h v h, v h ). (5.4) Set φ h = v h Γ, and decompose φ h into φ h = φ 0 + φe + e E Γ J m l [ Γ ij l=0 = φ (l) ij,, 0 + = φ (J) ij, ]. (5.5) Heeafte, φ 0, φe, φ (l) (J) ij,, 0 (l = 0,, J ) and φ ij, ae defined by Theoem 3.2. Define v 0 = E 0 φ 0, ve = Eeφe, v (l) ij,, 0 = E(l) ij,, 0 φ(l) ij,, 0 (l = 0,, J ) and ṽ(j) ij, = E(J) (J) ij, φ ij,. Moeove, we set fo each Ω k v k = {v h v 0 ve e E Γ J m l [ Γ ij l=0 = v (l) ij,,0 + = ṽ (J) ij, ]} Ω k. (5.6) Then, we have v k V p h (Ω k) by (5.5). It suffices to veify (5.4) fo the functions defined above. Fo convenience, set G(φ h ) = (B 0 v 0, v 0 ) + (Beve, ve) Ω e e E Γ + J m l [ (B (l) ij,, 0 v(l) ij,, 0, v(l) ij,, 0 ) + (B (J) ij, ṽ(j) ij,, ṽ(j) ij, )]. Γ ij l=0 = = By (4.7)-(4.0) and Lemma 5.-Lemma 5.4, we deduce G(φ h ) log(h/h) φ 0 2,Γ + + Γ ij [ Hee, we have used the fact that (F = Γ ij ) φ h H 2 (l) 00 (F ) e E Γ φe 2,Γ J m l φ (l) ij,, 0 2, Γ + l=0 = = (J) φ ij, 2, Γ]. (5.7) = φ h = 2, Ω i φ h 2, Ω, φ j h H 2 00 (F (l) ) (l = 0,, J; =,, m l ). Substituting (3.8) and (3.3) into (5.7), and note that J log(h/h), yields Using the tace theoem, we futhe get Besides, it follows by (5.6) that G(φ h ) log 3 (H/h) φ h 2,Γ G(φ h ) log 3 (H/h)(A h v h, v h ). (5.8) v k 2,Ω k v h 2,Ω k + v 0 2,Ω k + + J Γ ij [ e E Γ ve 2,Ω k J m l v (l) ij,, 0 2, Ω k + l=0 = = 28 ṽ (J) ij, 2, Ω k ].

29 This, togethe with (4.6)-(4.0), leads to n (B k v k, v k ) Ωk (A h v h, v h ) + J G(φ h ). k= Now, (5.4) is a diect consequence of the above inequality, togethe with (5.8). On the othe hand, one can deive the stength Cauchy-Schwaz inequality by using (4.6)-(4.0). Finally, we deduce (4.7) by the convegence theoy [28]. 6 Analysis fo the stability of the extension Ef This section is devoted to veification of the stability condition (4.2), which has been used in Lemma 4.2-Lemma 4.3. Since geneal sub-face f and geneal quasi-unifom meshes ae consideed, the analysis is a bit technical. Ou basic idea is to intoduce a suitable appoximate extension of Ef. This auxiliay extension is defined by positive integes, so that its stability can be veified moe easily by estimating two finite sums associated with the discete H semi-nom. 6. An auxiliay extension Fo a sub-face f Γ ij, let of, df, p and q be defined as in Subsection 4.. Without loss of geneality, we assume that h min{ pp : p N h Ω ij }. Fo a positive numbe x, let [x] denote the intege pat of x. Set nf = [ d f h ]. Fo any node p in Ω ij \f, define m p = [ pp h ] and n df p = [ ofq p q h ] (if p of). Fo convenience, set n p = nf if p = of. To undestand the meaning of the integes m p and n p moe intuitively, we image that the segments pp and p q ae divided into some smalle segments with the size h and h o f q espectively. Then, m p and n p can be viewed oughly d f as the numbes of the division points on the segments pp and p q espectively. In paticula, the position of a node p can be detemined by the two integes m p and n p oughly. By the definition of the sets Λ () f and Λ(2) f, it is easy to see that the positive integes m p and n p possess the popeties: Popety A. Fo each node p Λ () f Λ(2) f, we have m p n p nf; Popety B. Fo two integes and k satisfying k nf, thee ae at most O(nf) nodes p Λ () f Λ(2) f, such that these nodes p define the same m p = and n p = k; Popety C. Fo an intege k satisfying k nf, thee ae at most O(m p nf) nodes p Λ (2) f, such that these nodes p define the same n p = k. Fo the constant-like basis function φf = If 0, define the auxiliay extension φf(p), if p f, (E fφf)(p) mp = n f, if p Λ() f, (6.) mp n p, if p Λ (2) f, 0, othewise. Fo the veification of the stability (4.2), one needs to pove the following two inequalities (Ef E f)φf 2,Ω i df log(df/h) (6.2) 29