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1 This article appeared in a ournal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier s archiving and manuscript policies are encouraged to visit:

2 Comput. Methods Appl. Mech. Engrg. 197 (008) Contents lists available at ScienceDirect Comput. Methods Appl. Mech. Engrg. ournal homepage: Interior superconvergence in mortar and non-mortar mixed finite element methods on non-matching grids Gergina Pencheva a, *, Ivan Yotov b a Center for Subsurface Modeling, Institute for Computational Engineering and Sciences, The University of Texas at Austin, Austin, TX 7871, United States b Department of Mathematics, 301 Thackeray Hall, University of Pittsburgh, Pittsburgh, PA 1560, United States article info abstract Article history: Received 9 August 007 Received in revised form 14 February 008 Accepted 6 May 008 Available online 3 May 008 Keywords: Mixed finite element Mortar finite element Interior estimates Superconvergence Non-matching grids We establish interior velocity superconvergence estimates for mixed finite element approximations of second order elliptic problems on non-matching rectangular and quadrilateral grids. Both mortar and non-mortar methods for imposing the interface conditions are considered. In both cases it is shown that a discrete L -error in the velocity in a compactly contained subdomain away from the interfaces converges of order Oðh 1= Þ higher than the error in the whole domain. For the non-mortar method we also establish pressure superconvergence, which is needed in the velocity analysis. Numerical results are presented in confirmation of the theory. Ó 008 Elsevier B.V. All rights reserved. 1. Introduction The use of non-matching multiblock grids in discretizations of partial differential equations provides great flexibility in meshing highly irregular domains and accurately handling internal features, e.g., faults and geological layers in porous media. At the same time, special care has to be taken to impose properly interface continuity conditions. The computational error due to the non-matching grids affects the overall accuracy of the solution. In this paper we study how this interface error propagates into the interior of the subdomains. We consider the second order elliptic equation written as a first order system u ¼ Krp in X; ð1:1þ r u ¼ f in X; ð1:þ p ¼ g on ox; ð1:3þ where X R d, d = or 3, and K is a symmetric, uniformly positive definite tensor with L 1 ðxþ components. The above system models single-phase flow in porous media among many other applications. Here p is the pressure, u is the Darcy velocity, and K represents the rock permeability divided by the fluid viscosity. We assume that KðxÞ satisfies * Corresponding author. Tel.: ; fax: addresses: gergina@ices.utexas.edu (G. Pencheva), yotov@math.pitt.edu (I. Yotov). 8x X; k 0 n T n 6 n T KðxÞn 6 k 1 n T n 8n R d : ð1:4þ The Dirichlet boundary conditions are considered merely for simplicity of the presentation. Neumann boundary conditions u m ¼ g N and Robin boundary conditions ap þ u m ¼ g R, where m is the outward unit normal vector on ox, can also be considered. We assume that the problem (1.1) (1.3) is H -regular, i.e., there exists a positive constant C depending only on K and X such that kpk 6 Cðkf kþkgk 3=;oX Þ; ð1:5þ where H is the standard Hilbert space of functions having second order weak derivatives in L. Sufficient conditions for (1.5) are, for example, K C 0;1 ðxþ and X is convex or ox is smooth enough [0,]. We study the approximation of (1.1) (1.3) by mixed finite element methods on non-matching grids. Mixed methods are of interest due to their local mass conservation and high accuracy for both pressure and velocity, especially on structured grids. Multiblock discretizations allow for modeling highly irregular domains, while keeping the subdomain grids relatively simple. As a result, the use of mixed finite element subdomain discretizations can lead to accurate and efficient methods. Continuity of flux and pressure must be imposed on interfaces. We consider two approaches, a mortar and a non-mortar method. In the mortar mixed finite element method [1,1,34] a mortar finite element pressure Lagrange multiplier is introduced on the interfaces to impose weakly continuity of flux. If the mortar space contains polynomials of degree one order higher than the traces of /$ - see front matter Ó 008 Elsevier B.V. All rights reserved. doi: /.cma

3 4308 G. Pencheva, I. Yotov / Comput. Methods Appl. Mech. Engrg. 197 (008) subdomain velocities, the method is optimally convergent and even superconvergent in certain cases [1,34]. We refer the reader to [6,7,16,3,33] for some examples of applications of mortar spaces to other types of discretizations. In the non-mortar mixed finite element method [3], interface conditions are imposed without the use of a mortar space. On each subdomain, a partially hybridized mixed method is employed. Lagrange multiplier pressures are introduced on the element faces (or edges) on the subdomain interfaces C as in [4,11,19]. Since the grids are non-matching across C, the Lagrange multipliers are double-valued. Robin type conditions are imposed on C to couple the subdomain problems. The method has an advantage if adaptive local refinement techniques are to be used, since there is no mortar grid to refine. Such refinement could be difficult to implement in the mortar mixed method, since accuracy depends subtly on the relations between the mortar grid and the traces of the grids of the subdomain blocks [1], see the mortar condition (.8). In this paper we combine cutoff function and superconvergence techniques to establish interior error estimates for the velocity in mortar and non-mortar discretizations. We refer to [,15,17, 18,4] for relevant superconvergence results for mixed finite element methods on rectangular and quadrilateral elements. There are also examples of the use of cutoff functions in the context of mixed finite element methods. In [14], interior and superconvergence estimates are established in negative norms for linear and semi-linear elliptic problems. Interior estimates are established in [] for a cell-centered finite difference version of the mixed method, as well as in [31] for an enhanced velocity mixed finite element method. Here we establish interior superconvergence for the velocity in mortar and non-mortar mixed finite element methods. For both methods, if the grids are rectangular or mildly distorted quadrilateral, we show that the velocity converges of order Oðh 1= Þ higher in a compactly contained subdomain than in the whole domain. In the mortar case the superconvergence is the same as in single block discretizations with no boundary error. As a tool in the analysis we prove pressure superconvergence for the non-mortar method using a duality argument. Our analysis shows that the numerical error depends on the smoothness of the solution on every subdomain up to the interface. Numerical experiments confirm both the interior superconvergence for smooth solutions, as well as deterioration of interior convergence for singular solutions. The remainder of the paper is organized as follows. In the next section, we recall the mortar and non-mortar mixed finite element methods and their convergence properties. Section 3 is devoted to the interior error analysis for the mortar method. The non-mortar method is analyzed in Section 4. Numerical experiments confirming the theory are presented in Section 5.. Formulation of the methods and preliminaries We will use the following standard notation. For D R d, let ð; Þ and kk D be the L ðdþ inner product and norm, respectively. Denote the norm in the Hilbert space H s ðdþ by kk s;d. We may omit the subscript D if D ¼ X. Similar notation is used for S ox, with the exception that the L ðsþ inner product or duality pairing is denoted by h; i S. We denote by c and C generic positive constants, independent of the discretization parameter h. The velocity and pressure functional spaces for the mixed weak formulation of (1.1) (1.3) are defined as usual [11] to be V ¼ Hðdiv;XÞ ¼fv ðl ðxþþ d : r v L ðxþg; with norms kvk V ¼ðkvk þkr vk Þ 1= ; kwk W ¼kwk: W ¼ L ðxþ; A weak solution of (1.1) (1.3) is a pair u V; p W such that ðk 1 u; vþ ¼ðp; r vþ hg; v mi ox ; v Hðdiv;XÞ; ð:1þ ðr u; wþ ¼ðf ; wþ; w L ðxþ: ð:þ It is well known (see, e.g., [11,8]) that (.1) (.) has a unique solution. Let X be decomposed into non-overlapping subdomains (blocks) X i so that X ¼[ n X i and X i \ X ¼; for i 6¼. Let C i; ¼ ox i \ ox for i 6¼ ; C i;i ¼;; C ¼[ 16i<6n C i;, and C i ¼ ox i \ C ¼ ox i n ox denote interior block interfaces. We assume that the interfaces are flat. The blocks need not share complete faces, i.e., they need not form a conforming partition. Denote by V i ¼ Hðdiv;X i Þ; W i ¼ L ðx i Þ the subdomain functional spaces. To cast the problem (1.1) (1.3) in a multiblock form, we write on each block X i u ¼ Krp in X i ; ð:3þ r u ¼ f in X i ; ð:4þ p ¼ g on ox i \ ox; ð:5þ and on each interface C i; p i ¼ p ; u i m i þ u m ¼ 0; i.e., both the pressure and the flux are continuous across C i;. Here, m i is the outer unit normal to ox i and for any f defined on X, we denote both f Xk and its trace f Ck by f k. Let T h;i be a conforming, shape-regular, quasi-uniform finite element partition of X i ; 1 6 i 6 n, of maximal element diameter h i [13]. To simplify the presentation, we let h ¼ max 16i6n h i and analyze the methods in terms of this single value h. We allow for the possibility that the subdomain partitions T h;i and T h; do not align on C i;. Define T h ¼[ n T h;i and let V h;i W h;i V i W i be any of the usual mixed finite element spaces defined on T h;i (see [11, Section III.3]), the Raviart Thomas (RT) spaces [7,5], the Brezzi Douglas Marini (BDM) spaces [10], the Brezzi Douglas Fortin Marini (BDFM) spaces [9], the Brezzi Douglas Duràn Fortin (BDDF) spaces [8], or the Chen Douglas (CD) spaces [1]. We consider the above spaces on affine elements in R and R 3 as well as RT and BDFM spaces on convex quadrilateral elements in R. The order of the spaces is assumed to be the same on every subdomain. In the affine case, let V h;i contain the polynomials of degree k and W h;i contain the polynomials of degree l. For these spaces we have either l ¼ k or l ¼ k 1, with the former true for RT and BDFM spaces. The mixed finite element spaces on quadrilaterals are defined via a transformation to the reference square b E. For each element E, there exists a biection mapping F E : b E! E. Let DF E be the Jacobian matrix and let J E ¼detðDF E Þ. If b Vð b EÞ and c W ð b EÞ are the mixed finite element spaces on b E, then the spaces on E are defined via the transformations [9,11] v ¼ 1 DF E^v F 1 E ; w ¼ ^w F 1 E J : E The vector transformation is called the Piola transformation and preserves the normal components of the velocity vectors on the edges. It satisfies [11] ðr v; wþ E ¼ð^r ^v; ^wþ^e: ð:6þ The pressure superconvergence result for the non-mortar method will be shown for all of the above spaces on affine elements in R and R 3, as well as for the RT and BDFM spaces in R on h -parallelograms, see (.7). Interior superconvergence of the velocity in

4 G. Pencheva, I. Yotov / Comput. Methods Appl. Mech. Engrg. 197 (008) both methods will be shown for the RT and BDFM spaces on h - uniform quadrilateral grids, see (.7) (.8), and, if K is a diagonal tensor, on rectangular grids in R 3. Following the terminology from [18], a quadrilateral is called a h -parallelogram if it is a h -perturbation of a parallelogram: kðr r 1 Þ ðr 3 r 4 Þk 6 Ch ; ð:7þ where r i ; i ¼ 1;...; 4, are the vertices of the quadrilateral, see Fig. 1. A quadrilateral partition is called h -uniform, if each element is a h -parallelogram and any two adacent elements form a h -parallelogram, i.e., kðr r 1 Þ ðr 0 r0 1 Þk 6 Ch ; ð:8þ where r 0 i ; i ¼ 1;...; 4, are the vertices of the adacent element, see Fig. 1. We note that h -uniform grids can be constructed by uniform refinements of an initial quadrilateral grid or by smooth mapping of uniformly refined rectangular grids. The velocity and pressure mixed finite element spaces on X are defined as follows: V h ¼ n V h;i ; W h ¼ n W h;i : Note that the normal components of vectors in V h are continuous between elements within each block X i, but not across C. We introduce some proection operators needed in the analysis. For any of the standard mixed spaces there exists a proection operator P i : ðh e ðx i ÞÞ d \ V i! V h;i (for any e > 0), satisfying that for any q ðh e ðx i ÞÞ d \ V i, ðr ðp i q qþ; wþ ¼0; 8 w W h ; ð:9þ hðp i q qþm i ; v m i i Ci ¼ 0; 8v V h;i : ð:10þ Define P : n ððh ð i ÞÞ d \ V i Þ!V h such that P Xi ¼ P i. On affine elements, r V h;i ¼ W h;i : ð:11þ On quadrilateral elements, (.6) implies that on any element E r V h;i ðeþ ¼ 1 J E W h;i ðeþ: ð:1þ Since J E 6¼ constant, (.11) does not hold for quadrilaterals. Let b Q be the L ð b EÞ-orthogonal proection onto c W ð b EÞ, satisfying for any ^u L ð b EÞ, ð ^u b Q ^u; ^wþ ¼0; ^w c W ð b EÞ: Let Q h : L ðxþ!w h be the proection operator satisfying for any u L ðxþ, Q h u ¼ b Q ^u F 1 E on all E: It is easy to see that, due to (.6), ðu Q h u; r vþ ¼0; 8 v V h : ð:13þ For any / L ðcþ, let / i V h;i m i Ci be its L ðc i Þ-orthogonal proection satisfying h/ / i ; v m i i Ci ¼ 0; 8v V h;i : ð:14þ The proection operators have the following approximation properties: ku Q h uk 6 Ckuk t h t ; 0 6 t 6 l þ 1; ð:15þ kr ðq P i qþk Xi 6 Ckr qk t;xi h t ; 0 6 t 6 l þ 1; ð:16þ kq P i qk Xi 6 Ckqk r;xi h r ; 1 6 r 6 k þ 1; ð:17þ k/ / i k Ci; 6 Ck/k r;ci; h r ; 0 6 r 6 k þ 1; ð:18þ kðq P i qþm i k Ci; 6 Ckqk r;ci; h r ; 0 6 r 6 k þ 1: ð:19þ Bounds (.18) and (.19) are standard L -proection approximation results [13]; bounds (.15) (.17) can be found in [11,8] for affine elements and in [30,5] for h -parallelograms. In the analysis, we will also use the trace theorem kqk r;ci; 6 Ckqk rþ1=;xi ; r > 0; ð:0þ (see [0, Theorem ])..1. Mortar mixed finite element method If the solution ðu; pþ of (.1) (.) belongs to Hðdiv; XÞH 1 ðxþ, it is easy to see that it satisfies, for 1 6 i 6 n, ðk 1 u; vþ Xi ¼ðp; r vþ Xi hp; v m i i Ci hg; v m i i oxi nc i ; v V i ; ð:1þ ðr u; wþ Xi ¼ðf ; wþ Xi ; w W i : ð:þ Let T h;i; be a shape-regular, quasi-uniform affine finite element partition of C i; and let T C;h ¼[ 16i<6n T h;i;. Recall that k is associated with the degree of the polynomials in V h m. Denote by M h;i; L ðc i; Þ the mortar finite element space on C i; containing at least either the continuous or discontinuous piecewise polynomials of degree k þ 1onT h;i;. Let M h ¼ M h;i; 16i<6n be the mortar finite element space on C. In the mortar mixed finite element approximation of (.1) (.) we seek u h V h ; p h W h, and k h M h such that, for 1 6 i 6 n, ðk 1 u h ; vþ Xi ¼ðp h ; r vþ Xi hk h ; v m i i Ci hg; v m i i oxi nc i ; v V h;i ; ð:3þ ðr u h ; wþ Xi ¼ðf ; wþ Xi ; w W h;i ; ð:4þ hu h m i ; li Ci ¼ 0; l M h : ð:5þ Note that we have a standard mixed finite element method within each block X i and (.4) enforces local conservation over each element. It is clear from (.1) and (.3) that k h M h is an approximation to the pressure p on C. Eq. (.5) enforces weak flux continuity across C with respect to the mortar space M h. For the purpose of the analysis, it is convenient to eliminate k h from the mortar mixed method (.3) (.5) by restricting V h to the space of weakly continuous velocities ( ) V 0 h ¼ v V h : Xn hv i m i ; li Ci ¼ 0; 8l M h : Fig. 1. h -Uniform quadrilateral grid. Problem (.3) (.5) is equivalent to finding u h V 0 h p h W h such that and

5 4310 G. Pencheva, I. Yotov / Comput. Methods Appl. Mech. Engrg. 197 (008) ðk 1 u h ; vþ ¼ Xn ðp h ; r vþ Xi hg; v mi ox ; v V 0 h ; ð:6þ ðr u h ; wþ Xi ¼ðf ; wþ; w W h : ð:7þ Existence and uniqueness of a solution to (.3) (.5) are shown in [34,1] along with optimal convergence and superconvergence for both pressure and velocity under the following condition. Assumption.1. Assume that there exists a constant C independent of h such that klk Ci; 6 Cðkl i k Ci; þkl k Ci; Þ; l M h ; 1 6 i < 6 n: ð:8þ Remark.1. The condition (.8) imposes a limit on the number of mortar degrees of freedom and is easily satisfied in practice (see, e.g., [34,6]). Assuming (.8), the following superconvergence error estimates were shown in [1]: The Robin type interface conditions (.35) imply p i ¼ p ; u i m i þ u m ¼ 0; i.e., both the pressure and the flux are continuous across C i;. The non-mortar mixed finite element method is based on discretizing the subdomain variational equations (.1) (.) combined with a discrete weak imposition of the interface conditions (.35). Let K h;i to be the hybrid mixed finite element Lagrange multiplier space on C i [4,19], i.e., K h;i ¼ V h;i m i : In the non-mortar mixed finite element approximation of (.1) (.), we seek ð~u h ; ~p h ÞV h W h and ~ k h;i K h;i such that for 1 6 i 6 n, ðk 1 ~u h ; vþ Xi ¼ð~p h ; r vþ Xi h ~ k h;i ; v m i i Ci hg; v m i i oxi nc ; v V h;i; ð:36þ ðr ~u h ; wþ Xi ¼ðf ; wþ Xi ; w W h;i ; ð:37þ kpu u h k 6 C Xn ðkpk kþ5=;xi þkuk kþ3=;xi Þh kþ3= ; ð:9þ ha ~ k h;i ~u h;i m i ; l i i Ci ¼ Xn ¼1 ha ~ k h; þ ~u h; m ; l i i Ci; ; l i K h;i : ð:38þ kq h p p h k 6 C Xn ðkpk rþ;xi þkuk rþ1;xi þkruk rþ1;xi Þh rþ ; r ¼ minðk; lþ: ð:30þ Bound (.9) was shown only for RT elements on rectangular grids. In the analysis, we will also use the optimal velocity error estimate [1] kpu u h k 6 C Xn ðkpk kþ;xi þkuk kþ1;xi Þh kþ1 : ð:31þ The analysis in [1] is carried out for affine elements. It is easy to check that the estimates (.30) and (.31) also hold for h -parallelograms. The velocity superconvergence estimate (.9) relies on the proection error orthogonality ðk 1 ðp i u uþ; vþ Xi 6 Ch kþ kuk kþ;xi kvk Xi ; v V h;i ; ð:3þ which was shown in [15, Theorem 3.1], for RT and BDFM spaces on rectangular elements in R and R 3 and a diagonal tensor K. The results in [18] extend (.3) to h -uniform quadrilaterals and full tensor coefficients. In particular, it is shown in [18, Theorem 5.1], that ðk 1 ðp i u uþ; vþ Xi 6 Ch kþ ðkuk kþ;xi kvk Xi þkuk kþ1;xi kr vk Xi Þ; 8v V h;i ; v m i ¼ 0onoX i ; ð:33þ and ðk 1 ðp i u uþ; vþ Xi 6 Ch kþ3= ðkuk kþ3=;xi kvk Xi þkuk kþ1;xi kr vk Xi Þ; 8 v V h;i : ð:34þ Using (.34), it can be easily shown that (.9) also holds for h -uniform quadrilateral grids and full tensor coefficients. In this paper we employ (.33) to establish interior velocity superconvergence of order Oðh kþ Þ... Non-mortar mixed finite element method The non-mortar approach of [3] is based on using Robin type conditions for imposing continuity of pressures and fluxes across interfaces. To put the problem (.1) (.) in a multiblock form, choose a parameter a > 0 and write on each interface C i; ap i u i m i ¼ ap þ u m ; ap u m ¼ ap i þ u i m i : ð:35þ ¼1 We can replace (.38) with the condition that for 1 6 i 6 n hað ~ k h;i ~ k h; Þ; l i i Ci; ¼ Xn ¼1 h~u h;i m i þ ~u h; m ; l i i Ci; ; l i K h;i : ð:39þ Existence and uniqueness of a solution to (.36) (.38) are shown in [3] along with optimal convergence for both pressure and velocity: kp ~p h kþku ~u h kþ a X! 1= k ~ k h;i ~ k h; k C i; i; þ! 1 1= X k~u a h;i m i þ ~u h; m k C i; i; 6 C Xn ðkpk rþ3=;xi þkuk rþ3=;xi Þh rþ1 ; r ¼ minðk; lþ; ð:40þ ( ) 1= 6 C Xn kr ðu ~u h;i Þk X i kr uk lþ1;xi h lþ1 : 3. Interior estimates for the mortar mixed finite element method ð:41þ In this section, we establish interior superconvergence for the velocity in the mortar mixed finite element method. The results in the section are valid in the following cases. (A1) The mixed finite element spaces are either RT or BDFM. The grids are either h -uniform quadrilateral in R or rectangular in R 3. In the latter case, K is assumed to be a diagonal tensor. The interior error analysis is based on multiplying the test functions by appropriate smooth cutoff functions. We will make use of the following lemma. Lemma 3.1. If / H 1 ðxþ and v V h, then there exists a constant C independent of h such that kði PÞð/vÞk 6 Ckvkk/k 1 h: Proof. For any v V h, consider the functional l v ð/þ ¼/v Pð/vÞ:

6 G. Pencheva, I. Yotov / Comput. Methods Appl. Mech. Engrg. 197 (008) Since l v ð/þ ¼0 for all constant functions /, the statement of the lemma follows from an application of the Bramble Hilbert lemma [13]. h Let X 0 i be compactly contained in X i ; i ¼ 1;...; n and let X 0 ¼[ n X0 i. We prove the following interior velocity error estimates for the mortar mixed finite element method. Theorem 3.1. Assume (A1) and that (1.5) holds. Then, for any e > 0, there exists a constant C e dependent on the distance between X 0 and C, but independent of h such that kpu u h k X 0 6 C e ðkpk kþ;xi þkuk kþ;xi þkruk kþ1;xi Þh kþ e : Proof. Key ingredients in the proof of the theorem are the optimal velocity convergence (.31), the proection error orthogonality (.33), and the superconvergence for pressure (.30). The error equations for the mortar mixed finite element method are obtained by subtracting (.3) (.4) from (.1) (.): ðk 1 ðu u h Þ; vþ ¼ Xn ððp p h ; r vþ Xi hp; v m i i Ci Þ; v V 0 h ; ð3:1þ ðr ðu u h Þ; wþ Xi ¼ 0; w W h : ð3:þ Note that (3.), (.9), and either (.11) or (.1) imply that r ðpu u h Þ¼0: ð3:3þ For i ¼ 1;...; n, consider subdomain sequences X i ; ¼ 1; ;... such that X 0 i Xþ1 i X i X i: Let X ¼[ n X i. Let / þ1 C 1 0 ðx Þ be a cutoff function, / þ1 P 0; / þ1 1onX þ1, and / þ1 6 1onX. The constants that appear below may depend on k/ þ1 k 1;1;X. Note that, since / 1 on X, kvk X ¼k/ 1= vk X 6 k/ 1= vk X 1; which will be used repeatedly in our argument. We have, using (3.1) with v ¼ P/ þ1 ðpu u h Þ, ck/ 1= ðpu u þ1 hþk X 6 ðk 1 ðpu u h Þ; / þ1 ðpu u h ÞÞ X ¼ðK 1 ðpu u h Þ; ði PÞð/ þ1 ðpu u h ÞÞÞ X þðk 1 ðpu u h Þ; P/ þ1 ðpu u h ÞÞ X ¼ðK 1 ðpu u h Þ; ði PÞð/ þ1 ðpu u h ÞÞÞ X þðk 1 ðpu uþ; P/ þ1 ðpu u h ÞÞ X þ Xn ðp p h ; r ðp/ þ1 ðpu u h ÞÞÞ X i 6 CkPu u h k X kði PÞð/ þ1 ðpu u h ÞÞk X þ Ch kþ kuk kþ;x kp/ þ1 ðpu u h Þk X þ Ch kþ kuk kþ1;x kr P/ þ1 ðpu u h Þk X þðp p h ; r P/ þ1 ðpu u h ÞÞ X ; ð3:4þ where we have used either (.3) (for rectangular grids in R 3 )or (.33) (for h -uniform quadrilateral grids) in the last inequality. Using (.31) and Lemma 3.1, the first term on the right can be estimated as kpu u h k X kði PÞð/ þ1 ðpu u h ÞÞk X 6 C Xn ðkpk kþ;xi þkuk kþ1;xi Þh kþ1 kpu u h k X k/ þ1 k 1;X h 6 C Xn ðkpk kþ;xi þkuk kþ1;xi Þh kþ k/ 1= ðpu u h Þk X 1: ð3:5þ For the second term on the right in (3.4) we have kp/ þ1 ðpu u h Þk X 6 kði PÞ/ þ1 ðpu u h Þk X þk/ þ1 ðpu u h Þk X 6 ChkPu u h k X k/ þ1 k 1;X þk/ þ1 k 1;X kpu u h k X 6 Ck/ 1= ðpu u h Þk X 1; ð3:6þ using Lemma 3.1 for the second inequality. For the third term on the right in (3.4), using (.16) and (3.3), we have kr P/ þ1 ðpu u h Þk X 6 Ckr / þ1 ðpu u h Þk X ¼ Ckr/ þ1 ðpu u h Þk X 6 Ck/ þ1 k 1;1;X kpu u h k X 6 Ck/ 1= ðpu u h Þk X 1: ð3:7þ Similarly, using (.13), (.9) and (3.3), the last term on the right in (3.4) can be bounded as ðp p h ;rpð/ þ1 ðpu u h ÞÞÞ X ¼ðQ h p p h ;rpð/ þ1 ðpu u h ÞÞÞ X ¼ðQ h p p h ;r/ þ1 ðpu u h ÞÞ X 6CkQ h p p h k X k/ 1= ðpu u h Þk X 1 6C Xn using (.30) in the last inequality. Combining (3.4) (3.8), we get ðkpk kþ;xi þkuk kþ1;xi k/ 1= þ1 ðpu u hþk X 6 Ch kþ k/ 1= ðpu u h Þk 1= X 1 Xn þkruk kþ1;xi Þh kþ k/ 1= ðpu u h Þk X 1 ; ð3:8þ ðkpk kþ;xi þkuk kþ;xi þkr uk kþ1;xi Þ Ch kþ k/ 1= ðpu u h Þk 1= X 1 A 1= : Replacing þ 1 with in (3.9), we obtain k/ 1= ðpu u h Þk 1= 6 Ch kþ 4 k/ 1= ðpu u X 1 1 hþk 1=4 A 1=4 : X Multiplying (3.9) and (3.10) recurrently leads to k/ 1= ðpu u þ1 hþk X 6 Ch ðkþþð1 þ1 4 þþ A 1 þ1 4 þ 6 Ch kþ e A; where we take enough terms so that 1 þ 1 e þp 1. 4 kþ h! 1= ð3:9þ ð3:10þ 4. Interior estimates for the non-mortar mixed finite element method Similarly to the case of mortar mixed finite element method, subtracting Eqs. (.36) (.37) from (.1) (.) and using properties of the proections, we get the error equations for the non-mortar mixed finite element method ðk 1 ðu ~u h Þ; vþ Xi ¼ðQ h p ~p h ; r vþ Xi hp i ~ k h;i ; v m i i Ci ; v V h;i ; ð4:1þ

7 431 G. Pencheva, I. Yotov / Comput. Methods Appl. Mech. Engrg. 197 (008) ðr ðu ~u h Þ; wþ Xi ¼ 0; w W h;i : ð4:þ Here for clarity of notation we have denoted the trace of p on C i by p i, although, by the assumption p H 1 ðxþ, p has a single-valued trace on C. One of the key ingredients in the proof of Theorem 3.1, namely superconvergence for pressure, is missing in the analysis of [3]. Using a duality argument, we prove the following pressure superconvergence theorem. Theorem 4.1. Assume (1.5) and that any of the usual mixed spaces on affine elements or RT or BDFM on h -parallelograms are used. For the pressure ~p h of the non-mortar mixed method (.36) (.38), there exists a positive constant C, independent of h, such that kq h p ~p h k 6 C Xn ðkpk rþ3=;xi þkuk rþ3=;xi þkruk rþ1;xi Þh rþ3= ; r ¼ minðk; lþ: Proof. Let n H ðxþ be the solution of the auxiliary problem r Krn ¼ Q h p ~p h in X; n ¼ 0 on ox; and note that by (1.5) knk 6 CkQ h p ~p h k: ð4:3þ Let f ¼ Krn. Taking v ¼ P i f V h;i in (4.1) and using (.9), we have kq h p ~p h k X i ¼ðQ h p ~p h ; r P i fþ Xi ¼ðK 1 ðu ~u h Þ; P i f fþ Xi ðu ~u h ; rnþ Xi þhp i ~ k h;i ; P i f m i i Ci : Sum over i and use (.14) to get kq h p ~p h k ¼ Xn ðk 1 ðu ~u h Þ; P i f fþ Xi Xn ðu ~u h ; rnþ Xi þ Xn ¼ Xn þ Xn hp i ~ k h;i ; P i f m i i Ci ðk 1 ðu ~u h Þ; P i f fþ Xi þ Xn ðr ðu ~u h Þ; n Q h nþ Xi Xn hðu ~u h;i Þm i ; ni Ci hp i ~ k h;i ; f i m i i Ci ; ð4:4þ where we integrated by parts the second term and used (4.) and (.10). The first two terms on the right in (4.4) are easy to handle. Using (4.3) and the approximation properties (.17), (.15), we get ðk 1 ðu ~u h Þ; P i f fþ Xi þ Xn ðr ðu ~u h Þ; n Q h nþ Xi 6 C ku ~u h kþ Xn kr ðu ~u h Þk Xi!hkQ h p ~p h k: ð4:5þ Using the continuity of n and u, we rearrange the third term in (4.4) to obtain Xn hðu ~u h;i Þm i ; ni Ci ¼ 1 X hðu ~u h;i Þm i þðu ~u h; Þm ; ni Ci; i; h~u h;i m i þ ~u h; m ; n i i Ci; i; þ 1 X h~u h;i m i þ ~u h; m ; n n i i Ci; : i; ð4:6þ Bounding the second term on the right in (4.6) is straightforward: bounds (.18), (.0) and (4.3) give X h~u h;i m i þ ~u h; m ; n n i i Ci; 6 X! 1= k~u h;i m i þ ~u h; m k C i; i; i; X i; 6 C X i; h 6 C X i; X i kn n i k C i;! 1= k~u h;i m i þ ~u h; m k C i;! 1= knk 3=;X i! 1= k~u h;i m i þ ~u h; m k C i;! 1= hkq h p ~p h k: ð4:7þ To handle the first term on the right in (4.6), we take l i ¼ 1 n i in (.39), sum over i and rearrange to get 1 X h~u h;i m i þ ~u h; m ; n i i Ci; hað ~ k h;i ~ k h; Þ; n i i Ci; i; i; hað ~ k h;i ~ k h; Þ; n i n i Ci; i< hað ~ k h;i ~ k h; Þ; ðn i nþþðn n Þi Ci; i< 6 1 X ak ~ k h;i ~ k h; k Ci; i< ðkn i n i k Ci; þkn n k Ci; Þ 6 C X ak ~ k h;i ~ k h; k Ci; i< hðknk 3=;Xi þknk 3=;X Þ 6 C X ak ~ k h;i ~ k h; k Ci; hkq h p ~p h k; i< ð4:8þ where we used (.18), (.0) and (4.3) in the last inequality. It remains to bound the last term in (4.4). Using the continuity of f m and p, and rearranging terms, we obtain hp i ~ k h;i ; f i m i i Ci hp i ~ k h;i p þ ~ k h; ; f i m i i Ci; i; hp i p ~ k h;i þ ~ k h; ; ðf i P i f i Þm i i Ci; i; 1 X h ~ k h;i ~ k h; ; P i f i m i i Ci; i; þ 1 X hp i p ; P i f i m i i Ci; : i; ð4:9þ The first term in (4.9) can be bounded using (.18) (.0) and (4.3), hp i p ~ k h;i þ ~ k h; ; ðf i P i f i Þm i i Ci; 6 ðkp i p i k Ci; þkp p k Ci; þk ~ k h;i ~ k h; k Ci; Þkðf i P i f i Þm i k Ci; 6 C h kþ1 kp i k kþ3=;xi þ h kþ1 kp k kþ3=;x þk ~ k h;i ~ k h; k Ci; h 1= kfk 1;Xi 6 C h kþ1 kp i k kþ3=;xi þ h kþ1 kp k kþ3=;x þk ~ k h;i ~ k h; k Ci; h 1= kq h p ~p h k: ð4:10þ

8 G. Pencheva, I. Yotov / Comput. Methods Appl. Mech. Engrg. 197 (008) To handle the second term in (4.9), take l i ¼ a 1 P if i m i in (.39), sum over i, combine the two terms on C i; and use the continuity of f m to obtain 1 X h ~ k h;i ~ k h; ; P i f i m i i Ci; i; ¼ 1 X h~u a h;i m i þ ~u h; m ; P i f i m i i Ci; i; ¼ 1 X h~u a h;i m i þ ~u h; m ; P i f i m i þ P f m i Ci; i< hð~u a h;i m i þ ~u h; m Þ; ðf i P i f i Þm i þðf P f Þm i Ci; i< 6 1 X k~u a h;i m i þ ~u h; m k Ci; ðkðf i P i f i Þm i k Ci; i< þkðf P f Þm k Ci; Þ 6 C X i< 1 a k ~u h;i m i þ ~u h; m k Ci; h 1= kq h p ~p h k; ð4:11þ where we used (.19), (.0) and (4.3) in the last inequality. Finally, for the last term in (4.9), applying (.14) and using the continuity of p and f m, we get 1 X hp i p ; P i f i m i i Ci; i; hp i p ; P i f i m i i Ci; i; hp p ; P i f i m i þ P f m i Ci; i; hp p ; ðp i f i f i Þm i þðp f f Þm i Ci; i; 6 1 X kp p k Ci; ðkðf i P i f i Þm i k Ci; þkðf P f Þm k Ci; Þ i; 6 C X i; h kþ1 kpk kþ3=;x h 1= kq h p ~p h k; ð4:1þ using (.18) (.0) and (4.3) in the last inequality. A combination of (4.4) (4.1) and the use of (.40) and (.41) completes the proof. h We are now ready to establish an interior velocity error estimate for the non-mortar mixed finite element method. Theorem 4.. Assume (A1) and that (1.5) holds. Then, for any e > 0, there exist a constant C e dependent on the distance between X 0 and C, but independent of h such that The domain is divided into four (eight for Example 5.5) subdomains with interfaces along the x ¼ 1= and y ¼ 1= (and z ¼ 1= for Example 5.5) lines. Recall that optimal convergence for the solution to (.3) (.5) holds under the assumption that M h contains at least piecewise polynomials of degree k þ 1 [1], which in this case means piecewise linear functions. We test three types of methods on non-matching interfaces: 1 (continuous piecewise linear mortars), (discontinuous piecewise linear mortars), and 3 (Robin type interface conditions non-mortar mixed finite element method). In the fourth example we also test matching grids (denoted by mortar 0). In the numerical experiments we report the rates of convergence of the numerical solution (pressure and velocity) to the true solution. We compute the velocity error in the discrete L -norm v ¼ X X Eðv m e Þ ðm e Þ; ET h eoe where m e is the midpoint of edge (face) e. It is easy to see that c 1 kvk 6 v 6 c kvk 8 v V h and u Pu 6 Ch : Therefore, u u h 6 u Pu þ Pu u h 6 Cðh þkpu u h kþ and the superconvergence results from Theorems 3.1 and 4. hold for u u h as well. We also report the velocity error in the discrete L 1 -norm v 1 ¼ max e v m e ðm e Þ. The flux error is computed in the discrete L ðcþ-norm v m C ¼ X eðv m e Þ ðm e Þ: ec The pressure error is computed in the discrete L -norm w ¼ X ET h Ew ðm E Þ; where m E is the center of mass of element E. Since p Q h p 6 Ch ; p p h is also superconvergent. The convergence rates are established by running each test case on five (four for Examples 5.4 and 5.5) levels of grid refinement and computing a least squares fit to the error. The initial mesh on each block is uniform (shown on Fig. ) and the initial mortar grids on all interfaces are given in Table 1. The interior subdomains X 0 i are kpu ~u h k X 0 6 C e ðkpk kþ3=;xi þkuk kþ3=;xi þkr uk kþ1;xi Þh kþ3= e : The proof is similar to the proof of Theorem 3.1 and it is omitted. 5. Numerical results We present several numerical tests using the lowest order RT spaces (k ¼ 0) on rectangles and quadrilaterals to confirm the theoretical results of Sections 3 and 4. The first four examples are on the unit square and the fifth one is on the unit cube. The last two examples test quadrilateral grids on irregular domains obtained by a mapping of the unit square. Fig.. Initial grid. Table 1 Initial number of elements in mortar grids Mortar 1 3 Elements 3 3 1

9 4314 G. Pencheva, I. Yotov / Comput. Methods Appl. Mech. Engrg. 197 (008) Fig. 3. Solution and error (magnified) with mortar 1 for Example 5.1. chosen on the initial mesh to have a one-element border and are kept fixed during the mesh refinement. The boundary conditions are Dirichlet on the left and right edge and Neumann on the rest of the boundary. In all tests except the fourth one we solve problems with known analytical solutions. Example 5.1. We choose permeability I; 0 6 x < 1= K ¼ 10 I; 1= < x 6 1 Table Convergence rates for Example 5.1 Mortar ðu u h Þm C p p h u u h u u h 1 X Int. X Int Table 3 Convergence rates for Example 5. Mortar ðu u h Þm C p p h u u h u u h 1 X Int. X Int and the pressure ( pðx; yþ ¼ x y 3 þ cosðxyþ; 0 6 x < 1= y 3 þ cosð xþ9 yþ; 1= < x 6 1 xþ9 0 0 is chosen to be continuous and to have continuous normal flux at x ¼ 1=. Note that the solution is smooth in each subdomain. Plots of the computed solution and the numerical error for Example 5.1 using mortar 1 are shown in Fig. 3. The plots for all examples are on the third level of grid refinement, except for Example 5.5, where the plots are on the second level of refinement. As shown in Table, the interior velocity error is superconvergent of order Oðh Þ and most of the error occurs near the interfaces, as it can be seen from the flux error. The pressure error is also Oðh Þ-superconvergent. The results for the mortar method are as predicted by the theory and the results for the non-mortar method are approximately Oðh 1= Þ better than the theoretical results. Example 5.. In this example, we test a problem with a discontinuous full tensor 8 < 1 ; 0 6 x < 1= K ¼ 1 : I; 1= < x 6 1 and pressure xy; 0 6 x 6 1= pðx; yþ ¼ xy þðx 1=Þðy þ 1=Þ; 1= 6 x 6 1: Fig. 4. Computed pressure and velocity for Example 5..

10 G. Pencheva, I. Yotov / Comput. Methods Appl. Mech. Engrg. 197 (008) Fig. 5. Computed pressure and velocity error (magnified) for Example 5.. The convergence rates for Example 5. are given in Table 3 and confirm the theory. The computed pressure and velocity using the mortar and the non-mortar methods are shown in Fig. 4. Although the two solutions look the same, the velocity error along the interfaces is larger for the non-mortar method, as it can be seen in Fig. 5 where magnified numerical error is shown. Example 5.3. In the third example, we test non-uniform grids. We solve a problem with a known analytical solution pðx; yþ ¼x 3 y 4 þ x þ sinðxyþ cosðyþ and a diagonal tensor coefficient! K ¼ ðx þ 1Þ þ y 0 : 0 ðx þ 1Þ The initial grid is constructed from the grid on Fig. via the mapping Table 4 Convergence rates for Example 5.3 Mortar ðu u h Þm C p p h u u h u u h 1 X Int. X Int n þ 0:05 sinð4pnþ cosð1:5pnþ; 0 6 g 6 1=; x ¼ n þ 0:05 sinð4pnþ cosð0:3pnþ; 1= 6 g 6 1; g 0:03 sinð1pgþ; 0 6 n 6 1=; y ¼ g 0:05 sinð6pgþ cosð1:5pgþ; 1= 6 n 6 1; where n; g ½0; 1Š, and is refined uniformly for subsequent levels. For mortar we have started with a coarser mortar grid (two elements per mortar interface) in order to satisfy the stability condition (.8). The convergence rates are given in Table 4. As in the first two examples with uniform grids, the obtained convergence rates are in the accordance with the theory. Plots of the computed solution and the numerical error for Example 5.3 using mortar are shown in Fig. 6. It is evident from the first three examples that the error in the case of non-matching grids and piecewise smooth solutions occurs mainly along the interfaces and superconvergence is preserved in the interior. Example 5.4. In this example, we test a problem with a singularity due to a cross-point discontinuity in the permeability. In this test, analytical solution is not available. Instead, a fine grid solution is used to calculate the errors on all coarser grids. We test both matching and non-matching grids. The finest grid in the case of matching grids is The finest non-matching and mortar grids are shown in Table 6. The permeability tensor is K ¼ aðx; yþi, where Fig. 6. Solution and error (magnified) with mortar for Example 5.3.

11 4316 G. Pencheva, I. Yotov / Comput. Methods Appl. Mech. Engrg. 197 (008) Fig. 7. Solution and error (magnified) with mortar 1 for Example ; if x < 1=; y < 1= or x > 1=; y > 1= aðx; yþ ¼ 1; otherwise: L 1 -errors are not reported since the true velocity is not in L 1 ðxþ. As the results show (Fig. 7, Table 5), the error due to the strong singularity at the cross-point (1/, 1/) dominates the interface error and pollutes the solution in a large part of the domain. As a result, there is no superconvergence even in the interior. The rate of convergence for the interior velocity error is of order Oðh 3=4 Þ. In this case local grid refinement near this cross-point is needed to control the error [3]. Example 5.5. Next, we test a three dimensional problem with a full permeability tensor Table 5 Convergence rates for Example 5.4 Mortar ðu u h Þm C p p h u u h X Int Table 7 Convergence rates for Example 5.5 Mortar ðu u h Þm C p p h u u h u u h 1 X Int. X Int Table 8 Convergence rates for Example 5.6 Table 6 Finest grids for Example 5.4 A. Finest non-matching grids B. Mortar grids Mortar Elements Mortar ðu u h Þm C p p h u u h u u h 1 X Int. X Int Fig. 8. Solution and error (magnified) with mortar for Example 5.5.

12 0 1 x þðyþþ 0 cosðxyþ B K ¼ 0 z þ sinðyzþ A cosðxyþ sinðyzþ ðy þ 3Þ and pressure pðx; y; zþ ¼x 4 y 3 þ x þ yz þ cosðxyþ sinðzþ: Plots of the computed solution and the numerical error for Example 5.5 using mortar are shown in Fig. 8. The convergence rates are given in Table 7, again confirming the theoretical results. Example 5.6. In the last two examples, we test quadrilateral grids on irregular domains. We choose permeability and pressure as in Example 5.. In this example, the computational grids are constructed from the grid on Fig. and its uniform refinements via the mapping x ¼ n þ :06 cosðpnþ cosðpgþ; y ¼ g :1 cosðpnþ cosðpgþ; G. Pencheva, I. Yotov / Comput. Methods Appl. Mech. Engrg. 197 (008) where n; g ½0; 1Š. Note that the resulting quadrilateral grids are h - uniform. The convergence rates are given in Table 8. In calculating the rates, only the two finest levels were used due to inexact computation of the errors, which affects the results on coarse grids. Just as for the case of affine elements, the interior velocity is superconvergent of order Oðh Þ and most of the error occurs near the interfaces. The computed solution and error in pressure and velocity for Example 5.6 using mortar are shown in Figs. 9A and 10A. Example 5.7. In this example, the mapping used to generate the domain and the grids is x ¼ n þ :03 cosð3pnþ cosð3pgþ; y ¼ g :04 cosð3pnþ cosð3pgþ: The results for Example 5.7 are in Table 9, Figs. 9B and 10B. Comparing the numerical errors for Examples 5.6 and 5.7, we observe that although rougher mapping introduces larger error due to element distortion, interior velocity superconvergence is still obtained. Summarizing the test results in Section 5, we conclude that in smooth cases the numerical error due to the non-matching grids in both the mortar and the non-mortar methods is restricted to a small region around the interfaces and superconvergence is preserved in the interior. For singular solutions superconvergence is not observed, although the interior velocity error is better than the velocity error calculated over the whole domain. The results in this case indicate the need for a posteriori error estimates and adaptive mesh refinement near the points of singularity to increase the overall accuracy of the solution. Fig. 9. Computed pressure and velocity with mortar. Fig. 10. Computed pressure and velocity error (magnified) with mortar.

13 4318 G. Pencheva, I. Yotov / Comput. Methods Appl. Mech. Engrg. 197 (008) Table 9 Convergence rates for Example 5.7 Mortar ðu u h Þm C p p h u u h u u h 1 X Int. X Int Acknowledgement This work was partially supported by the NSF Grants DMS and DMS and the DOE Grant DE-FG0-04ER5618. References [1] T. Arbogast, L.C. Cowsar, M.F. Wheeler, I. Yotov, Mixed finite element methods on non-matching multiblock grids, SIAM J. Numer. Anal. 37 (000) [] T. Arbogast, M.F. Wheeler, I. Yotov, Mixed finite elements for elliptic problems with tensor coefficients as cell-centered finite differences, SIAM J. Numer. Anal. 34 (1997) [3] T. Arbogast, I. Yotov, A non-mortar mixed finite element method for elliptic problems on non-matching multiblock grids, Comput. Methods Appl. Mech. Engrg. 149 (1997) [4] D. Arnold, F. Brezzi, Mixed and non-conforming finite element methods: implementation, post-processing and error estimates, Math. Model. Numer. Anal. 19 (1985) [5] D.N. Arnold, D. Boffi, R.S. 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