c 1999 Society for Industrial and Applied Mathematics

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1 SIAM J. NUMER. ANAL. Vo. 36, No. 2, pp c 999 Society for Industria and Appied Mathematics OVERLAPPING NONMATCHING GRID MORTAR ELEMENT METHODS FOR ELLIPTIC PROBLEMS XIAO-CHUAN CAI, MAKSYMILIAN DRYJA, AND MARCUS SARKIS Abstract. In the first part of the paper, we introduce an overapping mortar finite eement method for soving two-dimensiona eiptic probems discretized on overapping nonmatching grids. We prove an optima error bound and estimate the condition numbers of certain overapping Schwarz preconditioned systems for the two-subdomain case. We show that the error bound is independent of the size of the overap and the ratio of the mesh parameters. In the second part, we introduce three additive Schwarz preconditioned conjugate gradient agorithms based on the trivia and harmonic extensions. We provide estimates for the spectra bounds on the condition numbers of the preconditioned operators. We show that athough the error bound is independent of the size of the overap, the condition number does depend on it. Numerica exampes are presented to support our theory. Key words. nonmatching grid, finite eement, mortar projection, overapping domain decomposition, eiptic equations, Schwarz preconditioner AMS subject cassifications. 65N30, 65F0 PII. S Introduction. The mortar eement method was first deveoped for the purpose of couping different discretizations in different nonoverapping subdomains. Severa studies have been carried out; see, e.g., [, 2, 3, 4, 5, 6, 7,, 2, 5, 6, 22, 25, 29, 30]. In this paper, we consider the case of overapping subdomains. We provide an optima error anaysis for the two-subdomain case, and we provide spectra bound estimations for the Schwarz preconditioned systems. The main advantage of nonmatching grid methods is that highy structured oca grids and corresponding fast sovers and software) can be used easiy. To preserve the goba accuracy of the discretization, the interpoation between the neighboring subdomains has to be sufficienty accurate. The mortar method provides one such interpoation scheme that passes the vaues of a function from one grid to another without osing accuracy, as wi be shown in this paper. It is somewhat surprising that the discretization error is independent of the overap as ong as a trivia requirement is satisfied; the overap is not smaer than the size of the coarser mesh. We aso show that the error is independent of the ratio of the mesh sizes. Another interesting finding is that arger overap can make the resuting inear system easier to precondition. We note that, independent of the deveopment of mortar based methods, overapping nonmatching grid techniques have been used Received by the editors June 27, 997; accepted for pubication in revised form) Apri 9, 998; pubished eectronicay March 2, Department of Computer Science, University of Coorado, Bouder, CO cai@cs. coorado.edu). The work of this author was partiay supported by NSF grants ASC and ECS and by NASA under contract NAS-9480 whie the author was in residence at the Institute for Computer Appications in Science and Engineering. Facuty of Math. Info. and Mech., Warsaw University, Warsaw, Poand dryja@mimuw.edu.p). The work of this author was partiay supported by NSF grant CCR and Poish Scientific grant 02/P03/95/09. Department of Mathematica Sciences, Worcester Poytechnic Institute, Worcester, MA 0609 markis@wpi.edu). The work of this author was partiay supported by NSF grant ASC , by NSF Grand Chaenges Appications Group grant ASC , and by NASA HPCC Group grant NAG

2 582 XIAO-CHUAN CAI, MAKSYMILIAN DRYJA, AND MARCUS SARKIS for more than 0 years by computationa engineers in many arge-scae simuations as a way to reduce the cost of grid generation. The methods are often referred to as the chimera methods or overset grid methods [3, 20, 26]. We are interested in soving the foowing eiptic variationa probem. Find u H 0 Ω), such that ) au,v)=fv), v H 0 Ω), where au, v) = u vdx and fv) = fv dx. Ω Ω Here fx) L 2 Ω) is a given function and Ω = Ω Ω 2 is an open poygona domain in R 2. We assume that both Ω and Ω 2 are open poygona domains and that the diameters of Ω, Ω, and Ω 2 are of order. We sha introduce two independent trianguations on Ω and Ω 2, respectivey, and a mortar eement method defined on the union of the two, generay nonmatching, trianguations. We assume that u satisfies the oca reguarity conditions u Ωi H +τi Ω i ) and 0 <τ i for i =, 2. No goba reguarity of u is assumed. As mentioned earier a ot of work has been done in the area of nonoverapping nonmatching grid methods. There are aso severa methods that use overapping nonmatching grid preconditioners for matrix probems obtained from nonoverapping discretization schemes; see [2, 5]. Some very interesting recent deveopment in using overapping nonmatching grid methods can be found, for exampe, in the papers of Kuznetsov [23], Bake [8], and Cai, Mathew, and Sarkis [0]. However, to the best of our knowedge, this is the first paper that provides an optima error anaysis for the overapping mortar eement method. To avoid unnecessary compications, we restrict our discussion to Poisson s equation with zero Dirichet boundary condition. The extension to the smooth variabe coefficient case is straightforward. The paper is organized as foows. In section 2, we introduce some notations. The mortar eement method and some impementation remarks are given in section 3. The anaysis of the method is provided in section 4. Severa technica emmas, used in section 4, are actuay introduced and proved in section 5. Section 6 reports severa numerica experiments that are used to verify the theory on the accuracy. Three preconditioning techniques are proposed and anayzed in section 7. Section 8 contains some numerica exampes supporting the theory of the preconditioning methods. A short concusion is given in section Mode cases and function spaces. In this paper, we sha focus on two mode cases that have different technica difficuties. The main theorem on accuracy hods for both cases; however, different proofs are needed. Most of our resuts can be extended to more genera cases. Case R: The union of Ω and Ω 2 is a rectanguar domain, as shown in Figure. Case L: The union of Ω and Ω 2 is an L-shaped domain, as shown in Figure 2. Before introducing the mortar eement method in Ω with nonmatching grids in the overapping subdomains, we need to define some notations. First, et γ i = Ω i Ω,i=, 2, be the interfaces. For Case R we define δ as the distance between the two interfaces, shown in Figure, and for Case L we assume δ = O).

3 OVERLAPPING MORTAR ELEMENT METHODS 583 y δ x 2 x Fig.. The subregions Ω i,i=, 2, are rectanges Ω i = i x y. i x,y are of O). δ is the size of the overap. Trianguations and finite eement spaces. Fori =, 2, et T hi = {K hi j,j =,...,M i} be a standard finite eement trianguation in Ω i ; see for exampe Figure. Here K hi j is a triange and h i the mesh size. M i is the tota number of trianges. We assume that they are shape reguar and quasi uniform; see Ciaret [4]. The two trianguations need not match in the overapping region. Let V hi V hi Ω i )bethe space of continuous piecewise inear functions on T hi which vanish on Ω Ω i.for each node x hi in T hi we denote by φ hi x) the usua basis function, i.e., φ hi x) V hi, and φ hi x) =ifx = x hi and zero at a the other nodes. We define the support of a basis function by suppφ hi ) suppx hi ) {x Ω i and φ hi x) 0}. Note that suppx hi ) is an open set. We aso need the space X h = {u,u 2 ) u i V hi,i=, 2}. We denote by V hi 0 a subspace of V hi containing a functions that vanish on Ω i. Trace spaces. We denote by V hi γ i ) the restriction of V hi on γ i. Let us denote by a i,a i 2,...,a i m i the nodes of T hi Ω i )on γ i, and aso denote by a i 0 and a i m i+ the two endpoints of γ i ; see Figure 2a) and Figure 3. We assume that if a i 0 or a i m i+) is anodeoft hi Ω i ), then a i 0 = a i or a i m i = a i m i+); see Figure and Figure 2a). It is important to note that for v i to beong to V hi, v i must vanish at a i and a i m i ; see Figure 3a) for an exampe of a function in V hi γ i ). Trivia extension operators. For any r i V hi γ i ), we define a function denoted by E i r i in V hi Ω i ) satisfying E i r i = r i at the nodes a i 2,a i 3,...,a i m i, and E i r i equas zero at the remaining nodes of T hi. Interface test function spaces. For i =, 2, Whi γ i ) denote the space of continuous piecewise inear functions on the grid a i 0,a i 2,...,a i m i,a i m i+, subject to the constraints that these continuous piecewise inear functions are constants in the intervas [a i 0,a i 2] and [a i m i,a i m i+]; see Figure 3b). Mortars, mortar spaces, and save nodes. The curve γ i has two sides. We refer to one of them as the mortar side and the other as the nonmortar side. In

4 584 XIAO-CHUAN CAI, MAKSYMILIAN DRYJA, AND MARCUS SARKIS 2 x y a 0 a a 2 a 3 a m a m = a m + γ 2 y γ 2 x a) b) Fig. 2. Case L. The union of Ω and Ω 2 is an L-shaped region. a 0 a a 2 a m a m a m+ a a 2 a m a m a m+ h γ a) Fig. 3. a) A function in the space V h γ ), which is the image of π. b) A test function in the space W h γ ). h γ b) most mortar eement methods see, e.g., [7]) the choice is rather arbitrary. In our case, we have ony one choice. For γ, we define the T h2 side as the mortar side and the T h side as the nonmortar side. On the nonmortar side, a finite eement space is defined by using the mortar projection given beow by 2). A simiar definition is used for γ 2. We define the mortar space V h2 γ ) resp., V h γ 2 )), as the restriction to the interface γ resp., γ 2 ) of the space V h2 resp., V h ). Among the points a i 0,a i,...,a i m i,a i m i+, as wi be seen ater, the vaues of the soution are known at a i 0,a i,a i m i, and a i m i+ through the given boundary conditions. We sha refer to the other points, a i 2,a i 3,...,a i m i, as the save nodes since their vaues are determined by the mortar projections to be defined beow. Mortar projections. The mortar projection π maps the space V h2 γ ) into V h γ ). Given a ϕ L 2 γ ), we set π ϕ) V h γ ) to zero in the intervas [a 0,a ] and [a m,a m +] and determine the vaues of π ϕ) at the save nodes a 2,a 3,...,a m by 2) ϕ π ϕ)ψds=0 ψ W h γ ). γ Simiary, we define the mortar projection π 2 on γ 2, which maps V h γ 2 )intov h2 γ 2 ).

5 OVERLAPPING MORTAR ELEMENT METHODS 585 The soution space. We define the soution space V h as foows: V h = { u,u 2 ) u i V hi,i=, 2, u γ = π u 2 γ ) and u 2 γ2 = π 2 u γ2 ) }. Before cosing this section, we need to make an important assumption under which the mortar projections are computabe. Assumption. Let a i k be a save node on γ i; then suppa i k) γ j = for i j and i, j =, 2. Remark 2.. For Case R, the assumption impies that δ max{h,h 2 }; otherwise the subdomains are not connected on the mesh eve. For Case L, it means that the two darkened regions in Figure 2b) do not intersect each other. Without this condition, the two mortar projections cannot be cacuated independenty. 3. Overapping mortar eement methods. In this section, we introduce the overapping mortar eement method and discuss some impementation issues, such as the construction of basis functions in V h. Our variationa probem associated with ) is defined by the foowing. Find u =u,u 2 ) V h, such that 3) a h u, v) =f h v) v =v,v 2 ) V h, where the weighted biinear form is defined as a h u, v) = u v dx + u v dx Ω \Ω 2 2 Ω Ω 2 + u 2 v 2 dx + u 2 v 2 dx 2 Ω Ω 2 Ω 2\Ω and f h v) = fv dx + fv dx Ω \Ω 2 2 Ω Ω 2 + fv 2 dx + fv 2 dx. 2 Ω Ω 2 Ω \Ω 2 The main motivation for defining the variationa probem this way is that the resuting stiffness matrix is symmetric. We wi show ater that the space V h is nonempty under Assumption. We remark that for matching overapping grids, by identifying the nodes that are in the overapping region, 3) reduces to the usua finite eement probem associated with ). In fact, ) is we defined for continuous functions, and in this case it is equivaent to 3). Since v i vanishes on part of Ω i, i =, 2, we can define a norm in X h by v 2 h = a h v, v). It is easy to see that the biinear form a h, ) is bounded in the sense that 4) a h u, v) u h v h u, v X h. For our estimate of the discretization error, we assume that u H +τ Ω ) H +τ2 Ω 2 ),

6 586 XIAO-CHUAN CAI, MAKSYMILIAN DRYJA, AND MARCUS SARKIS where 0 <τ i, for i =, 2. The main resut of the paper is summarized in the foowing theorem. Theorem 3.. Assume that Assumption is true. Then the exact soution u of ) and the mortar eement soution u of 3) satisfy u u h C 5) h τ u H +τ Ω) + h τ2 2 u H +τ 2 Ω2)), where C>0is a constant independent of h, h 2, h /h 2, h 2 /h, and δ. In the next few sections, we sha prove the theorem for both Case R and Case L, with sighty different techniques. We note that V h X h. The seection of basis functions in V h is not as trivia as in the usua finite eement case because the matching conditions have to be satisfied. As a resut of the mortar mapping, some of the basis functions, near the interfaces, are not oca functions, i.e., the support of the basis function covers a the eements that intersect the interface. Let Z i = {x hi, =,...,N hi 0 } be the set of noda points in Ω i, not incuding boundary or interface nodes. N hi 0 indicates the tota number of nodes in Ω i. For each x hi, reca that φ hi x) denotes the corresponding reguar finite eement basis function. Let Z i = { x hi, =,...,Ñ hi 0 } Z i be a subset of nodes such that suppx hi ) γ j for i j). For each x hi Z i, we define ψ hj = E j π j φ hi γj )), j i. Then, every function u =u,u 2 ) V h has a unique representation of the forms u = h u x )φ h x)+ h u 2 x 2 )ψ h x) x h Z x h 2 Z 2 and u 2 = h u 2 x 2 )φ h2 x)+ u x Z 2 x h Z x h 2 In summary, the basis functions have the forms h )ψ h2 x). and Ω : { φ h x), 0) if x h Z \ Z φ h x), ψ h x)) if x h Z { 0, φ h 2 Ω 2 : x)) if x h2 Z 2 \ Z 2 ψ h2 x), φ h2 x)) if x h2 Z 2. Note that the interface save nodes are not accounted for regarding the degree of freedoms. The tota degree of freedoms is N h 0 + N h2 0. The functions ψhi x) i =, 2) have to be precacuated by soving some sma inear systems of equations determined by the mortar projection. Two additiona inear systems need to be soved for finding the save vaues. The numbers of unknowns of these two inear systems are equa to the numbers of the save nodes on the interfaces. In the two-dimensiona cases that we consider, the inear systems are aways tridiagona, symmetric, and we conditioned due to the nature of the mortar projection. We note that two equivaent formuations for overapping nonmatching grids are given by Kuznetsov in [23]. One approach is based on a minimization principe and the other uses Lagrange mutipiers.

7 OVERLAPPING MORTAR ELEMENT METHODS Anaysis of the discretization error. To anayze the discretization error, we use the we-known second Strang s emma, in Strang and Fix [27], for the nonconforming situation. Let u and u be the soutions of ) and 3), respectivey. We have u u h inf v V h u v h + u v h ). Here and beow we use u to represent u Ω,u Ω2 ). Using the fact that u v 2 h = a h u v, u v) =a h u v, u v)+{f h u v) a h u,u v)} and 4), we obtain u v h u v h + f hu v) a h u,u v) u v h u f h w) a h u,w) v h + sup. 0 w V w h h Therefore, 6) u u h inf 2 u f h w) a h u,w) v h + sup. v V h 0 w V w h h In the rest of this paper, we sha refer to the first and second terms of the right-hand side of 6) as the best approximation error and the consistency error, respectivey. 4.. The best approximation error. Let us denote the subregion as the h union of a cosed simpices K j, where Kh j T h and K h j beongs to Ω Ω 2. Let us assume that Assumption hods; therefore, Ω h 2 is a nonempty connected open subregion. Let V h Ω h 2 ) denote the space of continuous piecewise inear functions on Ω h 2 that vanish on Ωh 2 \γ. Let H h 2 denote the discrete harmonic extension operator on V h Ω h 2 ) with boundary data on γ and zero data on Ω h 2 \γ. h2 Simiary, et us denote the subregion as the union of a cosed simpices K Ω h2 2 where K h2 j T h2 and K h2 j beongs to Ω 2 Ω. Let us assume that Assumption hods; therefore, Ω h2 2 is a nonempty connected open subregion. Let V h2 Ω h2 2 ) denote the space of continuous piecewise inear functions in Ω h2 2 Let H h2 2 denote the discrete harmonic extension operator in V h2 Ω h2 2 Ω h 2 j, which vanish on Ωh2 2 \γ 2. ) with boundary data on γ 2 and zero data on Ω h2 2 \γ 2. In the next emma, we prove that the best approximation error is optima. In the proof, we use severa technica emmas that wi be discussed in section 5. Lemma 4.. Assume Assumption hods. Then, for any u H +τi Ω i ), i =, 2, and 0 <τ,τ 2, there exists v =v,v 2 ) V h such that 7) and u v H Ω ) C ) h τ u H +τ Ω) + h τ2 2 u H +τ 2 Ω2) 8) u v 2 H Ω 2) C h τ u H +τ Ω) + h τ2 2 u H +τ 2 Ω2)). Here the constant C>0 is independent of h, h 2, h /h 2, h 2 /h, and δ.

8 588 XIAO-CHUAN CAI, MAKSYMILIAN DRYJA, AND MARCUS SARKIS Proof. We first construct w =w,w 2 ) X h. Let w i be a continuous piecewise inear function defined in Ω i by using the pointwise interpoation of u at the noda points of T hi. The standard interpoation theory [4] gives 9) u w i L 2 Ω i) + h i u w i H Ω i) Ch +τi i u H +τ i Ωi), 0 <τ i. Note, however, that w V h, in genera, since w i,i =, 2 do not vanish at the nodes {a i } and {a i m i }. Aso, w does not satisfy the matching conditions across the interfaces γ i,i=, 2. Let z i V hi be a continuous piecewise inear function that equas zero at the nodes a i and a i m i and equas w i at the remaining nodes of T hi. Thus, the piecewise inear function w i z i is equa to u a i )ata i and to u a i m i )ata i m i. Then by using Lemma 5.2 to be introduced in section 5), we obtain, for 0 <τ i, 0) and ) w i a i ) z i a i ) = u a i ) Ch τi i u H +τ i Ωi) w i a i m i ) z i a i m i ) = u a i m i ) Ch τi i u H +τ i Ωi). Since w i z i is equa to zero at a nodes of T hi except a i and a i m i, we can use 0), and ) to obtain, for 0 <τ i, 2) w i z i L 2 Ω i) + h i w i z i H Ω i) Ch +τi i u H +τ i Ωi), and consequenty, using a triange inequaity and 9), we obtain 3) u z i L2 Ω i) + h i u z i H Ω i) Ch +τi i u H +τ Ω i). Now z i V hi i =, 2), but z =z,z 2 ) V h because the matching conditions across the interfaces are not satisfied. To match the interface vaues, we need to further modify z i. Let and We define the function v =v,v 2 )as r = π z 2 γ )) z on γ r 2 = π 2 z γ 2 )) z 2 on γ 2. v i = z i + H hi 2 ri,i=, 2. Note that Assumption is used to guarantee the existence of H hi 2 ri. Note aso that H h 2 r resp., H h2 2 r2 ) vanishes on γ 2 resp., γ ). Since v i beongs to V hi Ω i ), for i =, 2, and they satisfy the matching conditions, v beongs to V h. We next show that v satisfies 7) and 8). By the triange inequaity 4) u v i H Ω i) u z i H Ω i) + H hi 2 ri H Ω i). The first term above has been estimated in 3). For the second term, we use Lemma 5.0 to obtain H hi 2 ri H Ω i) C r i 2 + ) 5) H /2 00 γi) δ ri 2 L 2 γ i).

9 OVERLAPPING MORTAR ELEMENT METHODS 589 We bound r L2 γ ), and simiary r 2 L2 γ 2), as foows: r L2 γ ) = π z 2 z L2 γ ) = π z 2 π z L2 γ ) π z 2 π u L2 γ ) + π z π u L2 γ ). A consequence of the L 2 stabiity of Lemma 5.4 is that r L2 γ ) 6 z 2 u L2 γ ) +6 z u L2 γ ). Using Assumption, we have that z 2 = w 2 on γ. Then z 2 u L 2 γ ) = w 2 u L 2 γ ). According to the standard estimate for pointwise interpoation, we get, for 0 <τ 2, that 6) w 2 u L2 γ ) Ch /2+τ2 2 u H +τ 2 Ω2). Thus, we have obtained 7) π z 2 π u L2 γ ) Ch /2+τ2 2 u H +τ 2 Ω2), 0 <τ 2. We aso have π z π u L 2 γ ) 6 z u L 2 γ ) and therefore, by using a triange inequaity, z u L2 γ ) w u L2 γ ) + u a u )φ h a + a L 2 m )φ h L a. γ ) m 2 γ ) Using the above estimate, together with 0), ), and 2), we arrive at π z π u L2 γ ) Ch /2+τ u H +τ Ω), 0 <τ. This impies 8) r i L 2 γ i) C 2 i= h /2+τi i u H +τ i Ωi), i =, 2. We next bound r /2 H γ), and simiary r2 /2 00 H 00 γ2). We use the H/2 00 stabiity of Lemma 5.4 to obtain r H /2 00 γ) π z 2 π u H /2 00 γ) + π z π u H /2 00 γ) C z 2 u /2 H 00 γ) +6 z u /2 H γ). 00 Now with 3) we get 9) 2 r i /2 H 00 γi) C h τi i u H +τ i Ωi), i =, 2. i= Finay 7) and 8) foow immediatey from 4), 5), 8), 9), and the fact that δ is arger than max{h,h 2 }.

10 590 XIAO-CHUAN CAI, MAKSYMILIAN DRYJA, AND MARCUS SARKIS 4.2. The consistency error. The consistency error can be estimated rather easiy. For a smooth u, by using Green s formua and that u = f in the L 2 sense, we obtain f h w) a h u,w)= f + u )w dx u Ω 2 γ n w )ds + u 2 γ2 n w )ds + f + u )w 2 dx u Ω 2 2 γ2 n w 2)ds + u 2 γ n w 2)ds = 2 γ γ2 u n [w]ds = 2 γ u n w 2 w )ds + 2 γ2 u n w w 2 )ds, where u n denotes the norma derivative of u with the unit vector n pointing to the outside of Ω Ω 2. Later, we use the density argument Grisvard [9]) to estimate f h w) a h u,w) for any u H Ω). We summarize the resut in the foowing emma. Lemma 4.2. Let u H +τi Ω i ), 0 τ i,i =, 2. Then there exists a constant C>0 independent of δ, h i, and u such that sup 0 w V h 0 u γ γ 2 n [w]ds C h τ w u H +τ Ω) + h τ2 2 u H +τ 2 Ω2)). h Proof. We derive a bound for the consistency error on γ. The bound on γ 2 can be obtained in a simiar way. Let w =w,w 2 ) V h ;wehave u γ n w 2 w )ds = u n w 2 π w 2 )ds, and by using the definition of the mortar mapping 2), we aso have ψ W h γ ) u γ ) n w 2 π w 2 )ds u = n ψ w 2 π w 2 )ds γ u n ψ w 2 π w 2 H /2 γ ) [H /2 γ )] u n ψ w2 H /2 γ ) + w H /2 γ )). [H /2 γ )] Appying the trace theorem for w, we deduce that { u γ n w 2 w )ds C w u h inf n ψ γ ψ W h γ ) [H /2 γ )] With the hep of Lemma 5. or Lemma 4. of Bernardi, Maday, and Patera [7]), we obtain u γ n w 2 w )ds Chτ w u h Ch τ n w h u H +τ Ω). H /2+τ γ) }.

11 OVERLAPPING MORTAR ELEMENT METHODS Technica emmas. In this section we discuss severa technica estimates. We formuate and prove some of the emmas in a way that is more genera than needed in this paper since we beieve their appicabiities go beyond this paper. The proof of the foowing emma can be found in Bernardi, Maday, and Patera [7], athough their definition of the mortar mapping is sighty different from ours for Case L because of the two extra intervas [a i 0,a i ] and [a i m i,a i m i+]. Their proof aso hods here because the engths of the intervas [a i 0,a i 2] and [a i m i,a i m i+] are Oh i ); we do not incude the proof here. Lemma 5.. Let π i be the orthogona projection from L 2 γ i ) onto W hi γ i ). Then, for any 0 τ i, the foowing estimate hods for any v H τi γ i ): v π i v L2 γ i) + h /2 i v π i v [H /2 γ i)] Chτi i v H τ i γ i). As a consequence, { } /2+τ inf v ψ [H ψ W /2 γ i)] Ch i i v H τ i γi). hi γ i) Here C>0 is independent of h i. The next emma is usefu ony for Case L. Let us restrict our arguments to Ω ; a simiar argument appies for Ω 2. Reca that in the definition of the finite eement space V h Ω ), we insist that the functions vanish at two interior points a and a m, which is a bit unusua in the cassica finite eement theory. Due to the foowing emma, we show that the interior zero points do not affect the second-order or + τ i - order) accuracy of the overa discretization. Lemma 5.2. Let Ω be a bounded open subset of R 2 with a piecewise C 0, boundary Ω. Assume that the aspect ratio and the size of Ω are both O). Letν Ω be a C, differentiabe Lipschitz) curve with end points A and B. Aso et η ν Ω be an open nonempty connected curve with end points A and x 0. Then for any u H +τ Ω ), 0 <τ, that vanishes on Ω, we have 20) ux) Cd τ x u H +τ Ω) x ν. Here d x is the arc distance of the point x to η aong the curve ν. The constant C>0 does not depend on u, x 0, and x but in genera depends on the Lipschitz constant of Ω. Proof. If x η, then ux) = 0 and 20) hods triviay. Let us assume that x ν \ η. Let zx) be a point in the interior of η such that dzx),x 0 ) dx, x 0 )=d x. We sha first assume that u is a smooth function and then pass it to any functions in H +τ Ω ) using the cassica density argument; see, e.g., Grisvard [9] or Lions and Magenes [24]. Now et u C Ω ); then ux) =uzx)) + x zx) Since uzx)) = 0 and u s) =0ons η, wehave ux) = x x 0 u s)ds. u s)ds.

12 592 XIAO-CHUAN CAI, MAKSYMILIAN DRYJA, AND MARCUS SARKIS Using the Schwarz inequaity, we have x 2) ux) u s) ds d /2 x x 0 With the fundamenta theorem of cacuus, we have u s) =u zx)) + and using that u s) =0ons η, weget ux) = x s x 0 zx) s zx) u H ν). u t)dt, u t)dtds. By using the fact that u y) =0,y η, the Schwarz inequaity, and that dx 0,zx)) dx, x 0 ), we obtain 22) ux) Cdx, x 0 ) 3/2 u H 2 ν). We obtain the estimate in H +τ ν) by interpoating the H ν) estimate 2) and the H 2 ν) estimate 22) Lions and Magenes [24]). Thus, for 0 τ, 23) ux) Cd /2+τ x u H +τ ν). With the usua density argument, the above estimate hods for any u H +τ ν). Finay, to obtain 20) from 23), we consider two cases, /2 τ and 0 <τ /2, separatey. For /2 τ, we use the trace theorem for C 0, differentiabe Lipschitz) curve see Theorem.5.2. of Grisvard [9]), which gives ux) Cd τ x u H /2+τ ν) Cd τ x u H +τ Ω). For 0 <τ /2, it is known that the continuous function space is embedded into H /2+τ Ω ). Using that u vanishes on η, we can use the Brambe Hibert emma and scaing arguments to obtain, for 0 <τ /2, ux) Cd τ x u H +τ Ω) u H +τ Ω ). The ast arguments can be found in detai in the proof of Theorem 3.3 in [3]. Remark 5.3. We remark that we use the above emma by taking x 0 = a 0 or x 0 = a m +) and ν as an edge of an eement K h j of T h Ω ) that contains a 0 and a. The emma is usefu ony when a 0 a, and therefore using the definition of a 0 and a ) a 0 beongs to the interior of ν. We next show the boundness of the mortar projection in two different norms. Since the mortar projection is, in some sense, cose to the reguar L 2 projection, the L 2 bound is rather easy to obtain. It is a bit invoved to obtain its H /2 00 bound. Lemma 5.4. The mortar mapping π i is bounded in L 2 γ i ), i.e., 24) π i w L2 γ i) 6 w L2 γ i) w L 2 γ i ), and π i is aso bounded in H /2 00 γ i), i.e., 25) π i w H /2 00 γi) C w H /2 00 γi) w H /2 00 γ i),

13 OVERLAPPING MORTAR ELEMENT METHODS 593 where the constant C>0 is independent of h, h 2, h /h 2, h 2 /h, and δ. Proof. Let us consider the proof for π. The proof for π 2 is simiar. Using 2) and taking ψ, here denoted by v, which equas to π w at the noda points a 2,a 3,...,a m, we obtain Using simpe cacuations, we have π w 2 L 2 γ ) π w, v) L2 γ ) =w, v) L2 γ ). v 2 L 2 γ ) 6 π w 2 L 2 γ ), and 24) foows easiy. We next estimate the H /2 00 bound. Let w H0 γ ). By the triange inequaity and then the inverse inequaity, we have 26) π w 2 H /2 00 γ) C h π w Q h w 2 L 2 γ ) + Q h w H /2 00 γ) Here Q h : V h2 γ ) V h γ ) is the usua orthogona L 2 projection. Note that π Q h w = Q h w. Therefore, using 24) we have ). 27) π w Q h w 2 L 2 γ ) = π w π Q h w 2 L 2 γ ) C w Q h w L 2 γ ). The next step is to bound w Q h w L 2 γ ). Now we foow the proofs of Theorems 3.2 and 3.4 of Brambe and Xu [9]. Let us denote by I h the usua noda vaue interpoant on the grid a 0,a,a 2,...,a m,a m +. The interpoator is we defined in H γ ). Let us denote by φ a i the standard basis functions associated to the continuous piecewise inear functions on the grid a 0,a,a 2,...,a m,a m +. It is easy to see that beongs to V h γ ). Therefore, w = I h w wa )φ a wa m )φ a m w Q h w L2 γ ) w w L2 γ ) w I h w L 2 γ ) + wa )φ a L 2 γ ) + wa m )φ a m L 2 γ ). Since I h w is we defined for w H γ ), by using a we-known resut of Ciaret [4] we obtain w I h w L2 γ ) Ch w H γ ). Using that w vanishes at a 0 and a m +, wehave and then obtain wa ) Ch /2 w H γ ) and wa m ) Ch /2 w H γ ) 28) w Q h w L 2 γ ) Ch w H γ ). Using that Q h is a L 2 projection, we have w Q h w L2 γ ) 2 w L2 γ ). Then by the interpoation procedure we obtain 29) w Q h w L2 γ ) Ch /2 w H /2 00 γ).

14 594 XIAO-CHUAN CAI, MAKSYMILIAN DRYJA, AND MARCUS SARKIS The next step is to show that 30) Q h w H γ ) C w H γ ). Let w 0 = w on [a 0,a ] and [a m,a m +], and w 0 = wa )φ a x) on[a,a 2] and w 0 = wa m )φ a m x) on[a m,a m ], and zero at the remaining points of γ. Hence, ) Q h w 2 H γ 2 ) Q h w w 0 ) 2 H γ + Q ) h w 0 2 H γ ). By using an inverse inequaity, the L 2 stabiity resut 24), and the definition of w 0,wehave Q h w 0 2 H γ C ) h 2 Q h w 0 2 L 2 γ C ) h 2 w 0 2 L 2 γ C ) h 2 w 2 L 2 a 0,a ) ) + wa )φ a 2 L 2 a,a 2 ) + wa m )φ a m 2 L 2 a m,a m ) + w 2 L 2 a m,a m + ) C w 2 H γ ). In the ast inequaity, we use 2), which hods for functions w that vanish at a 0 and a m +. Note that Q h w w 0 )= Q h w w 0 ), where Q h is the standard L 2 projection in the space of piecewise inear functions defined on the grids a,a 2,...,a m and vanish at the end points a and a m. Hence by using standard resuts of the L 2 projection and some previous arguments we obtain 30) by Q h w w 0 ) 2 H γ ) C w w 0 2 H γ ) C w 2 H γ ) +C h 2 ) wa )φ a 2 L 2 a,a 2 ) + wa m )φ a m 2 L 2 a m,a m ) C w 2 H γ. ) We then use 30), the L 2 stabiity of Q h, and an interpoation procedure to obtain 3) Q h w 2 H /2 00 γ) C w H /2 00 γ). The inequaity 25) foows from 3), 29), 27), and 26). To simpify the discussion of the next emma we assume that Ω =0, ) 0, ) is a unit square with sides parae to the coordinate axes. The resut of the foowing emma can be extended to any Lipschitz regions by using the techniques deveoped in, e.g., Nečas [2]. Let the x-coordinate of γ equa. Let Γ δ Ω be the set of points that is within a distance δ of γ and define ζ = Γ δ Ω. Thus the x-coordinate of ζ equas δ). Lemma 5.5. There exists a constant C>0 independent of δ, such that 32) and 33) w 2 L 2 ζ) C w 2 L 2 γ ) + δ w 2 H Γ δ ) ) w 2 L 2 ζ) C δ w 2 H Γ δ ) + δ w 2 L 2 Γ δ ) )

15 OVERLAPPING MORTAR ELEMENT METHODS 595 hod for any w H Ω ). Proof. By using the fundamenta theorem of cacuus we have w w δ, y) =w,y) s, y)ds. δ x Squaring both sides and taking the integra in y from 0 to, we obtain 0 0 w δ, y)) 2 dy w,y)) 2 dy Now using the Schwarz inequaity on the ast term, w δ, y)) 2 dy 2 w,y)) 2 dy +2 δ δ δ ) 2 w s, y)ds dy. x ) 2 w s, y) ds) dy, x and 32) foows. To prove 33), we note that for x δ, ), x w w δ, y) =wx, y) s, y)ds, δ x which impies, by squaring both sides and using the Schwarz inequaity, that ) 2 w w δ, y)) 2 2 wx, y) 2 + δ s, y) ds). x The proof of 33) is now had by integrating this inequaity over δ, ) 0, ). Remark 5.6. A simiar estimate pays a very important roe in the study of the optima convergence of the overapping Schwarz methods with sma overap; see Dryja and Widund [7]. The next two emmas are devoted to Case R. For a given overap δ, we introduce a finite eement trianguation of size Oδ) onω. More precisey, we et T δ Ω )bea trianguation of Ω, which may or may not be nested with T h Ω ). We assume the trianguation is quasi uniform with size Oδ) and V δ Ω ) is the space of continuous piecewise inear functions on the trianguation T δ Ω ). We denote by γ δ the set of noda points of T δ Ω ) beonging to γ. Foowing Dryja, Sarkis, and Widund [8], we define an interpoation operator Iδ M : V h Ω ) V δ Ω ) as foows. Definition 5.7. Given w V h Ω ), define w δ = Iδ M w V δ Ω ) by the vaues of w δ at two types of nodes of T δ Ω ): i) For an interior noda point P T δ Ω )\γ, δ et τ P T δ Ω ) be a triange with P as one of its vertices. We define w δ P ) as the average of w over τ P, i.e., τ P wdx/ τ P dx. ii) For a boundary noda point P γ, δ et τ P T δ Ω ) be a triange with P as one of its vertices and having an edge on γ. We define w δ P ) as the average of w over τ P γ, i.e., the ine integra τ P γ wds/ τ P γ ds. Lemma 5.8. There exists a constant C>0, independent of δ and h, such that δ 34) I I M δ )w L2 Ω ) Cδ w H Ω ), 35) I M δ w H Ω ) C w H Ω ),

16 596 XIAO-CHUAN CAI, MAKSYMILIAN DRYJA, AND MARCUS SARKIS and 36) I M δ w L2 γ ) C w L2 γ ) hod for any w V h γ ). Remark 5.9. A proof can be found in the paper of Dryja, Sarkis, and Widund [8]. The interpoation operator Iδ M is used ony as part of the proof of the next emma, not in the impementation of any of the agorithms proposed in this paper. For the next emma, et us assume that ζ is aigned with the h -grid, and et H δ be the h -discrete harmonic extension operator in V h Γ δ ) with boundary data on γ and zero data on Γ δ \γ. Aso, et H be the h -discrete harmonic extension operator in V h Ω ) with boundary data on γ and zero data on Ω \γ. Lemma 5.0. There exists a constant C>0 independent of δ and h, such that Hw δ 2 H Γ δ ) w C 2 + ) 37) H /2 00 γ) δ w 2 L 2 γ ) for any w V h γ ). Proof. Using a triange inequaity, we have H δ w 2 H Γ δ ) 2 Hδ w I M δ w) 2 H Γ δ ) +2 Hδ I M δ w 2 H Γ δ ) =2I +2I 2. Let θ δ be a smooth function with vaues equa to one on γ and to zero on Ω \Γ δ. Let I h be the usua pointwise piecewise inear continuous interpoation operator. Using the fact that the discrete harmonic extension has minima energy, I I h θ δ H w I M δ H w)) 2 H Ω ) C H w Iδ M H w 2 H Ω + ) ) δ 2 H w Iδ M H w 2 L 2 Ω ). In the ast inequaity, we used the standard estimate as in the additive Schwarz theory see, e.g., [7]). Finay we use 34) and 35) to obtain I C H w 2 H Ω ) C w 2 H /2 γ). 00 Using again that the discrete harmonic extension has minima energy, and estimating 36), we obtain I 2 C Iδ M w) 2 x k ) C δ w 2 L 2 γ. ) x k γ δ The proof of the emma foows immediatey. Remark 5.. This emma is used ony for Case R. 6. Numerica experiments: Accuracy. To support the accuracy theory deveoped in the ast few sections, we conduct some numerica experiments. We consider ony Case R, and the probem domain is shown in Figure. In a tests, we assume that the exact soution u has the form u x, y) = π )) sinπx) + sin 2 x sinπy)

17 OVERLAPPING MORTAR ELEMENT METHODS 597 Tabe The initia grid on Ω is 6 5 and on Ω 2 is 5 4. The eement sizes are h =0.2 and h 2 =0.25. δ =0.45. In row, the number in parentheses is the ratio with the number in row. The ratio indicates the order of the accuracy of the discretization. L 2 L H L e) = D = 2.274D ) 3.754D ) ) ) = D ) 9.469D ) ) ) =3.480D ) 2.375D ) ) ) = D ) 5.945D ) 8.927D ) ) = D ).486D ) 4.463D ) 5.429D ) Tabe 2 We fix the refinement to =5, i.e., h =0.2/32 and h 2 =0.25/32. The grids are 60 + ovp) 60 and 28 + ovp) 28. L 2 L H L e) ovp = 9.59D-05.45D D D-02 ovp =2 9.58D-05.45D D D-02 ovp =4 9.70D-05.47D D D-02 ovp =8 9.90D-05.42D D D-02 ovp = D D D D-02 ovp = D D D D-02 and Ω = 0, 2) 0, ). We denote Ω 0 =0, ) 0, ), Ω 0 2 =, 2) 0, ), and the computed soution u =u,u 2 ) V h. Let I hi be the pointwise piecewise inear interpoation operator in T hi. The error that we report in this section is defined by e =e,e 2 )=I h u u,i h2 u u 2 ). Our theory appies ony to the H norm, but three discrete norms L 2, L, and H are used to measure the numerica error. More precisey, we use e L 2 Ω) = e 2 L 2 Ω 0 ) + e 2 2 L 2 Ω 0). 2 Simiary, we can define e H Ω). e L Ω) is given as e L Ω) = max{ e L Ω ), e L Ω 2)}. The refinement is done by simpy cutting each triange into four equa trianges. We use to denote the eve of refinement. In the first test case, we take h and h 2 cose to each other. We choose Ω = 0,.2) 0, ) and Ω 2 =0.75, 2) 0, ). The overapping size is fixed to δ =0.45. The initia mesh i.e., = 0) sizes are h =0.2 and h 2 =0.25, which transate to two nonmatching grids of 6 5 and 5 4. The resuts are summarized in Tabe. Five eves of uniform refinements are performed. One can see ceary that the method is of first order in H Ω) and of second order in L 2 Ω). We next examine the dependence on the overap. We fix the mesh sizes at h = 0.2/32 and h 2 =0.25/32, i.e., the refinement eve = 5. Let ovp be an integer denoting the number of eements in the x direction in the overapping region; we et ovp go from to 32. The resuts can be found in Tabe 2. As predicted in Theorem 3., the accuracy is independent of the overap.

18 598 XIAO-CHUAN CAI, MAKSYMILIAN DRYJA, AND MARCUS SARKIS Tabe 3 We fix the overap δ = The initia grid is 6 5 and 5 4. Shown is the error on Ω and Ω 2 when we refine both grids uniformy with different eve of refinement denoted by Ω and Ω2, respectivey. L 2 L H L e) error in Ω Ω =3, Ω2 = D D Ω =4, Ω2 = 8.26D ) 2.238D ) ) ) Ω =5, Ω2 =2 2.9D ) 6.77D ) ) ) error in Ω 2 Ω =3, Ω2 = D D Ω =4, Ω2 =.294D ) 2.596D ) ) ) Ω =5, Ω2 =2 3.30D-033.9) 6.709D ) 8.646D ) ) Instead of using the same eve of refinement in both subdomains, we experiment with a different eve of refinement denoted by Ω and Ω2. We aso measure the error separatey in Ω and Ω 2. We start with the same initia mesh 6 5 and 5 4) and refine three times in each subdomain with eves equa to Ω =3, 4, 5, and Ω2 =0,, 2. The resuts are provided in Tabe Additive Schwarz preconditioners. The inear system of equations corresponding to 3) is usuay arge, sparse, symmetric positive definite, and i conditioned. Preconditioning is necessary if iterative methods are used to sove it. In this section, we introduce severa additive Schwarz preconditioners. A good introduction on the abstract additive Schwarz method ASM) and its theory can be found in the book by Smith, Bjørstad, and Gropp [28]. The key eement of the abstract ASM theory is the introduction of a bounded decomposition of the finite eement soution space V h. Three such decompositions wi be discussed in this section. Some numerica resuts are given at the end to support our theory. 7.. An additive Schwarz method based on the harmonic extension ASHE). We first introduce a method that uses discrete harmonic extensions in the overapping region. The subspace decomposition is given by V h = I V + I 2 V 2, V = V h 0 Ω ), V 2 = V h2 0 Ω 2), where the interpoation operator I : V h 0 Ω ) V h Ω) is given as foows. For v V h 0 Ω ), we define I v V h Ω) by v in Ω interior, zero on γ ), I v = π 2 v on γ 2, H h2 2 π 2v in Ω 2, and the interpoation operator I 2 : V h2 0 Ω 2) V h Ω) is given as foows. For v 2 V h2 0 Ω 2), we define I 2 v 2 V h Ω) by I 2 v 2 = v 2 in Ω 2 interior, zero on γ 2 ), π v 2 on γ, H h 2 π v 2 in Ω. Let the biinear forms b i u i,v i ):V hi 0 Ω i) V hi 0 Ω i) R,i=, 2, be defined by 38) b i u i,v i )=a i u i,v i ) u i v i dx. Ω i

19 OVERLAPPING MORTAR ELEMENT METHODS 599 The subspace projection operator T i : V h Ω) V hi 0 Ω i),i=, 2, satisfies b i T i u, v) =a h u, I i v) v V hi 0 Ω i). Now we define the operator T i = I i Ti : V h Ω) V h Ω) and et T = T + T 2. To anayze the spectra condition of the operator T, we use the abstract ASM theory. The foowing emma is a sighty modified version of the abstract ASM emma in Smith, Bjørstad, and Gropp [28] for two overapping subregions with no coarse space. Lemma 7.. Suppose the foowing three assumptions hod: i) There exists a constant C 0 such that u V h Ω) there exists a decomposition u = 2 i= I iu i,u i V hi 0 Ω i), with 2 b i u i,u i ) C0a 2 h u, u). i= ii) There exist constants ɛ ij,i,j =, 2, such that a h I i u i, I j u j ) ɛ ij a h I i u i, I i u i ) /2 a h I j u j, I j u j ) /2 u i V hi 0 Ω i) u j V hj 0 Ω j). iii) There exists a constant ω such that a h I i u i, I i u i ) ωb i u i,u i ) u i V hi 0 Ω i),i=, 2. Then, T is invertibe, a h Tu,v)=a h u, T v) u, v V h Ω), and 39) C 2 0 a hu, u) a h Tu,u) ρe)ω)a h u, u) u V h Ω). Here ρe) is the spectra radius of E, which is a 2 2 matrix made of {ɛ ij }. We estimate the condition number of T in the next theorem. Both Case R and Case L are considered. For Case R, we define the overapping size δ as usua, and for Case L, we assume that δ = O). Theorem 7.2. Assume that Assumption hods. Then cδa h u, u) a h Tu,u) Ca h u, u) u V h Ω), where c>0 and C>0 are constants independent of h i and δ. Therefore if the overap is sufficienty arge, i.e., δ = O), the preconditioner is optima. Proof. We foow the abstract theory stated in Lemma 7.. We need ony to verify the three assumptions. Assumption i). Given v =v,v 2 ) V h Ω), we define u i V hi 0 Ω i) as foows: and u = v H h 2 v = v H h 2 π v 2 ) in Ω u 2 = v 2 H h2 2 v 2 = v 2 H h2 2 π 2v ) in Ω 2.

20 600 XIAO-CHUAN CAI, MAKSYMILIAN DRYJA, AND MARCUS SARKIS It is easy to check that u i V hi 0 Ω i) and that v = I u + I 2 u 2, since I u + I 2 u 2 = { v H h 2 v + H h 2 π v 2 = v in Ω H h2 2 π 2v + v 2 H h2 2 v 2 = v 2 in Ω 2. For i =, 2wehave ) 40) a i u i,u i ) 2 a i v i,v i )+a i H hi 2 v i, H hi 2 v i) C δ a iv i,v i ). To obtain the ast inequaity, we use Lemma 5.0 and the standard trace theorem H hi 2 v i 2 H Ω h i 2 ) C v i 2 + ) H /2 00 γi) δ v i 2 L 2 γ i) C δ a iv i,v i ). Note that the above inequaity hods for Case L with δ = O). From 40), we obtain C 2 0 = C/δ, since b u,u )+b 2 u 2,u 2 ) C δ a hu, u). Assumption ii). It is easy to see that ρe) 2. Assumption iii). We prove for i =. Let u V h 0 Ω ). Then ) a h I u, I u ) 2a u,u )+2a 2 H h2 2 π 2u ), H h2 2 π 2u ). To bound the second term, we again use Lemma 5.0, which impies that H h2 2 π 2u ) 2 H Ω h 2 2 ) C π 2 u 2 + ) H /2 00 γ2) δ π 2u 2 L 2 γ 2). To bound π 2 u /2 H 00 γ2), we appy the H/2 00 stabiity resut of Lemma 5.4 π 2 u 2 H /2 00 γ2) C u 2 H /2 00 γ2) Ca u,u ). To bound π 2 u L 2 γ 2), we use the L 2 stabiity resut of Lemma 5.4, π 2 u 2 L 2 γ 2) C u 2 L 2 γ 2), and we use the fact that u vanishes on γ and by Lemma 5.5 we have u 2 L 2 γ 2) Cδ b u,u ). Therefore ω = C, which appears in the above inequaity. Remark 7.3. We remark that if the overap is sufficienty arge, i.e., δ = O), then the agorithm is optima in the sense that the convergence rate is independent of the mesh parameters h and h 2. The arge overap condition is satisfied automaticay for Case L.

21 OVERLAPPING MORTAR ELEMENT METHODS An additive Schwarz method based on the trivia extension ASTE). We propose another additive Schwarz method in which the harmonic extension operator used in the previous subsection is repaced by a trivia zero extension. This method is computationay cheaper and easier to impement. Let us reca the definition of the trivia extension operators. For i =, 2 et E i r i : V hi γ i ) V hi Ω i )be the zero extension of r i to Ω i ; i.e., E i r i = r i at the nodes a i 2,a i 3,...,a i m i and E i r i equas zero at the remaining nodes of T hi. The subspace decomposition is given by V h = ÎV + Î2V 2, V = V h 0 Ω ), V 2 = V h2 0 Ω 2), where the interpoation operator Î : V h 0 Ω ) V h Ω) is given as foows. For v V h 0 Ω ), we define Îv V h Ω) by v in Ω, Î v = π 2 v on γ 2, E 2 π 2 v in Ω 2, and the interpoation operator Î2 : V h2 0 Ω 2) V h Ω) is given as foows. For v 2 V h2 0 Ω 2), we define Î2v 2 V h Ω) by v 2 in Ω 2, Î 2 v 2 = π v 2 on γ, E π v 2 in Ω. The biinear forms b i u i,v i ):V hi 0 Ω i) V hi 0 Ω i) R,,i =, 2, are defined the same as in 38). We define the projection operator ˆT i : V h Ω) V hi 0 Ω i),i=, 2, by b i ˆT i u, v) =a h u, Îiv) v V hi 0 Ω i). Now we define the operator T i = Îi ˆT i : V h Ω) V h Ω) and et T = T + T 2. The spectra bounds of T are estimated in the foowing theorem. Again, for Case L, we assume δ = O). Theorem 7.4. Assume that Assumption hods and et h = min{h,h 2 }. Then cha h u, u) a h Tu,u) C δ h a hu, u) u V h Ω), where c>0 and C>0 are constants independent of h i and δ. Proof. We ony need to verify the assumptions in Lemma 7.. Assumption i). Given v =v,v 2 ) V h Ω), we define u i V hi 0 Ω i) as foows: and u = v E v = v E π v 2 ) in Ω u 2 = v 2 E 2 v 2 = v 2 E 2 π 2 v ) in Ω 2. It is easy to check that u i V hi 0 Ω i) and that u = Îu + Î2u 2. It is straightforward to show that b i u i,u i ) C h i a i v i,v i ) C h i a h v, v)

22 602 XIAO-CHUAN CAI, MAKSYMILIAN DRYJA, AND MARCUS SARKIS and therefore C 2 0 = C/h. Assumption ii). It is easy to see that ρe) 2. Assumption iii). We discuss ony the case i =. Let u V h 0 Ω ). Then a h Îu, Îu ) 2a u,u )+a 2 E 2 π 2 u ), E 2 π 2 u ))). Using an inverse inequaity and the L 2 stabiity resut of Lemma 5.4, we obtain a 2 E 2 π 2 u ), E 2 π 2 u )) C h 2 π 2 u 2 L 2 γ 2) C h 2 u 2 L 2 γ 2). Reca the fact that u =0onγ, and using Lemma 5.5 we have u 2 L 2 γ 2) Cδ u 2 H Ω Ω 2). Note that for Case L, δ can be repaced by. Therefore, Simiary, we can get a h Îu, Îu ) C δ h 2 b u,u ). a h Î2u 2, Î2u 2 ) C δ h b 2 u 2,u 2 ). Thus, we can take ω = Cδ/h. Remark 7.5. The agorithm is not optima, and both ower and upper bounds are dependent on h and the overapping size δ. However, the agorithm is easy to impement. A sighty improved version of the agorithm is given in the next subsection. A comparison with ASHE is given in section 8. Remark 7.6. The upper bound depends on δ in a rather bad way, i.e., it increases when the overap increases. This aso shows up in the numerica exampes. Remark 7.7. We note, however, that the ower bound for Case R can be improved from Ch to Ch/ δ) for arge overap. For the proof we use 32) to obtain E v 2 H Ω C ) v 2 L h 2 γ C ) v 2 2 L h 2 γ C δ ) v 2 H Ω\Ω h ) A method based on a modified trivia extension ASTE). Both the upper and the ower bounds of ASTE depend on the mesh parameters. Here we propose a modification of the biinear form b i, ) and as a resut the upper bound becomes independent of the mesh parameters. We assume the subspace decomposition is the same as in the previous subsection. Here we modify the biinear forms; i.e., b i u i,v i ):V hi 0 Ω i) V hi 0 Ω i) R,,i =, 2, are now defined by b u,u ) + h h 2 ) a u,u )+ h h 2 x D h 2 u 2 x) and b 2 u 2,u 2 ) + h ) 2 a 2 u 2,u 2 )+ h 2 h h x D h 2 u 2 2x).

23 OVERLAPPING MORTAR ELEMENT METHODS 603 Here D hi j i j) denotes the set of mesh points x in the trianguation T hi, such that suppx) γ j. We define the projection operator T i : V h Ω) V hi 0 Ω i),i=, 2, by b i T i u, v) =a h u, Îiv) v V hi 0 Ω i). Now we define the operator T i = Îi T i : V h Ω) V h Ω) and et T = T + T 2. Theorem 7.8. Assume that Assumption hods. Then c + ) a h u, u) a h Tu,u) Ca h u, u) u V h Ω), h h 2 where c>0 and C>0 are constants independent of h i and δ. Proof. We exam the assumptions in Lemma 7.. Assumption i). Given v =v,v 2 ) V h Ω) we define u i V hi 0 Ω i) as foows: and We have b u,u ) 2 u = v E v = v E π v 2 ) in Ω u 2 = v 2 E 2 v 2 = v 2 E 2 π 2 v ) in Ω 2. + h ) a v,v )+a E v, E v )) + h h 2 C + h ) a v,v )+ ) v 2 L h 2 h 2 γ ) + C h 2 v 2 L 2 γ 2 ) + v 2 L 2 γ + 2 ) ), v h x) 2 2 D h 2 where γ 2 + and γ 2 are the ines parae to γ 2 and contain the noda points of D h 2. Using the standard trace theorem, we have b u,u ) C + ) a v,v ). h h 2 And simiary b 2 u 2,u 2 ) C + ) a 2 v 2,v 2 ). h h 2 Adding these estimates, we get b u,u )+b 2 u 2,u 2 ) C + ) a h u, u). h h 2 Therefore, C 2 0 = C h + h 2 ).

24 604 XIAO-CHUAN CAI, MAKSYMILIAN DRYJA, AND MARCUS SARKIS Tabe 4 A comparison of four methods in terms of the iteration numbers and condition numbers, given in parentheses. The initia grids are 6 5 and 5 4. The overap is fixed at δ =0.45. is the eve of refinement. no prec ASHE ASTE ASTE = ) 43.0) 73.7) 9 3.8) = ) 42.2) 226.5) 25.5) = ) 42.6) 284.8) 269.4) = ) 42.5) ) 37.3) = ) 32.5) 548.4) 3933.) = ) 32.5) ) ) Assumption ii). ρe) 2. Assumption iii). For u V h 0 Ω ), and using the L 2 stabiity of Lemma 5.4, a h Îu, Îu ) 2a u,u )+C h 2 u 2 L 2 γ 2). Now we use inequaity 33) for a strip D h 2 of width 2h, i.e., u 2 L 2 γ C 2) h u 2 + ) H D h u 2 ) 2 h L 2 D h, 2 ) to obtain a h Îu, Îu ) C Simiary, we have + h h 2 ) a u,u )+ h u 2 h x) = Cb u,u ). 2 D h 2 a h Î2u, Î2u 2 ) Cb 2 u 2,u 2 ). Thus, we obtain ω = C. Remark 7.9. Note that the bounds that appear in the emma are independent of the overapping parameter δ, even for Case R. Numerica exampes given in the next section indeed show that increasing overap does not decrease the number of iterations. 8. Numerica resuts: Preconditioning. In this section, we present some numerica resuts concerning the convergence rate of the preconditioned conjugate gradient PCG) methods. We are particuary interested in the dependence of the agorithms on the mesh parameters h and h 2 and the overapping size δ. A tests are for Case R. In Tabe 4, we present the number of PCG iterations and the condition number of the preconditioned system for each of the three agorithms, pus the case when no preconditioner is used. We stop the iteration when the initia preconditioned residua is reduced by a factor of 0 2. The initia grids are 6 5 and 5 4, and the grids are refined simutaneousy for up to = 5 times. The overapping size is fixed at δ =0.45. It can be seen ceary that the number of iterations for ASHE stays as a constant; however, a other methods have some dependence on the refinement eve. The modified method ASTE is consideraby better than ASTE.

25 OVERLAPPING MORTAR ELEMENT METHODS 605 Tabe 5 Verifying the overapping size. The mesh sizes are h =0.2/2 5 and h 2 =0.25/2 5. The actua meshes are 60 + ovp) 60 and 28 + ovp) 28. Note that ovp =32is the same as δ =0.45. no prec ASHE ASTE ASTE ovp = 75448) ) 66.0) 440.0) ovp = ) ) ) ) ovp = ) 222.6) ) ) ovp = ) 76.) ) ) ovp = ) 53.3) ) ) ovp = ) 32.5) ) ) In the second set of tests, we fix the mesh sizes and vary the overapping parameter δ. As predicted by our theory, ASHE gets better when the overap becomes arger. The other two preconditioners do not share this property. The resuts can be found in Tabe 5. We shoud mention that athough ASTE and ASTE do not perform as we as ASHE they sti have practica vaue since they are much easier to impement. 9. Concuding remarks. In the first part of the paper, we introduced a mortar finite eement method defined on overapping nonmatching grids. An optima accuracy theory is provided for the two-subdomain cases. When a geometrica condition is satisfied we prove that the accuracy is independent of the overap, as we as the ratio of the subdomain mesh sizes. In the second part of the paper, we studied three additive overapping Schwarz preconditioning techniques. One of the preconditioners, based on the oca harmonic extension, is optima in the sense that the convergence rate of the corresponding PCG method is independent of the mesh parameters h and h 2. Much more work needs to be done in the area of overapping mortar eement methods, such as extending the methods and theory to the case when more than two subdomains overap and to three-dimensiona probems. REFERENCES [] G. Abdouaev, Y. Achdou, J. Hontand, Y. Kuznetsov, O. Pironneau, and C. Prud homme, Non-matching grids for fuids, in Tenth Internationa Conference on Domain Decomposition Methods for Partia Differentia Equations, J. Mande, C. Farhat, and X.-C. Cai, eds., AMS, Providence, RI, 998. [2] Y. Achdou, Y. Maday, and O. Widund, Méthod itérative de sous-structuration pour es ééments avec joints, C. R. Acad. Sci. Paris, Ser. I, ), pp [3] Y. Achdou, Y. Maday, and O. Widund, Iterative substructuring preconditioners for mortar eement methods in two dimensions, TR735, Department of Computer Science, Courant Institute of Mathematica Sciences, New York University, New York, 997. [4] Y. Achdou and Y. Kuznetsov, Agorithms for the mortar eement method, in Domain Decomposition Methods in Science and Engineering, R. Gowinski, J. Periaux, Z.-C. Shi, and O. Widund, eds., John Wiey, New York, 997. [5] Y. Achdou, J. Hontand, and O. Pironneau, A mortar eement method for fuids, in Domain Decomposition Methods in Science and Engineering, R. Gowinski, J. Periaux, Z.-C. Shi, and O. Widund, eds., John Wiey, New York, 997. [6] F. Ben Begacem, The mortar finite eement method with Lagrange mutipiers, Numer. Math., to appear. [7] C. Bernardi, Y. Maday, and A. Patera, A new nonconforming approach to domain decomposition: The mortar eement method, in Coege de France Seminar, H. Brezis and J. Lions, eds., Pitman, Boston, MA, 990. [8] D. Bake, Appication of unstructured grid domain decomposition techniques to overset grids, in Proceedings of the Eighth SIAM Conference on Parae Processing for Scientific Computing, SIAM, Phiadephia, 997.