A Mixed Finite Volume Method for Elliptic Problems

Transcription

1 A Mixed Finite Volume Method for Elliptic Problems Ilya. Mishev and Qian-Yong Chen ExxonMobil Upstream Research Company P.O. Box 2189, Houston, TX Abstract We derive a novel finite volume method for the elliptic equation, using the framework of mixed finite element methods to discretize the pressure and velocities on two different grids (covolumes), triangular (tetrahedral) mesh and control volume mesh. The new discretization is defined for tensor diffusion coefficient and well suited for heterogeneous media. When the control volumes are created by connecting the center of gravity of each triangle to the midpoints of its edges, we show that the discretization is stable and first order accurate for both scalar and vector unknowns. Key words; finite volume methods, cell-centered finite differences, error estimates AMS subject classification: 65N06, 65N12, 65N15. Postdoctoral associate at Institute for Mathematics and its Applications, University of Minnesota. Partially funded by ExxonMobil Upstream Research Company. Current address: epartment of Mathematics and Statistics, University of Massachusetts Amherst 1

2 1 Introduction Physical diffusion process is usually modeled by an elliptic equation. The heterogeneous diffusion tensor due to the heterogeneous anisotropic media imposes great challenges to the numerical schemes. There are several highly desirable properties of the discretization beyond the classical stability and accuracy, such as local mass conservation, discrete maximum principle, etc, that are crucial in capturing the important details of the solutions of complicated (especially nonlinear) problems on relatively coarse grids. In this paper we derive a method that is locally mass conservative and it can be applied to a general finite volume grid constructed from any given triangular or tetrahedral mesh. The first version belongs to the so called class of K methods. When the control volume is formed by centers of gravity and edge midpoints of the triangles, we show first order accuracy for the pressure gradient and the velocity. The included numerical results confirm the theoretical estimates. We only mention the second version that is one of the K 1 methods. This version could be more suitable for simulation of fluid flow in porous media where harmonic averaging has proven to be very useful for problems with discontinuous coefficients when the control volumes are aligned to the discontinuities. The investigation of the K 1 method is beyond the scope of this paper. Our computational experiments show that when we use Voronoi boxes and elaunay triangles the resulting matrices from both versions are M-matrices which is in agreement with known results for Finite Element Methods [38]. Our discretization is similar to the Finite Volume Element (FVE) method, but it also produces direct approximation of the velocity which is not the case for FVE method. This can be very beneficial in some applications. In the proposed method, the velocity unknowns are eliminated and the resulting linear system for the pressure can be solved with the available solvers. We used ILU with fill-in type of preconditioners [31], but Algebraic Multigrid is also applicable. The estimated computational cost on triangular or tetrahedral mesh will be two to three times less than the cost for solving the same problem with the standard mixed finite 2

3 element methods. There have been extensive research on developing numerical schemes for the partial differential equations of interest on a given general grid. The classical finite difference schemes are only applicable to structured grids. The mimetic finite difference schemes proposed by Shashkov [33] are very promising in terms of dealing with highly distorted grids. The convergence and super convergence [6, 7] are also established for smooth problems on smooth meshes by rewriting it into the form of mixed finite element methods. But the computational cost is an issue. Finite volume methods overcome most of the restrictions of finite difference schemes, and they are usually locally mass conservative. There have been a significant advance in the theory of the finite volume methods applied to diffusion equations with scalar coefficient on unstructured meshes [2, 18, 22, 24, 30]. Several methods for handling tensor coefficient have been proposed [37, 15, 1]. Recently the convergence is established for one of such methods, the multi-point flux approximation, on quadrilateral grids under the assumption of smooth mesh [20]. But a comprehensive theory for general mesh is still not available yet. Control volume finite element methods, often called finite volume element methods are another worthy alternative [5, 16, 11, 32, 17, 25]. They are applicable on flexible grids and are locally mass conservative, but do not preserve the symmetry of the differential operator, and do not produce direct approximation of the velocity field. Mixed finite elements, since their introduction [28], have proven to be robust and give very accurate approximations. In some cases it is possible to reduce the system of coupled equations for the pressure and velocity to a system only for the pressure [4, 3], which is much easier to solve. Unfortunately some restrictions apply. There is considerable advance in designing methods for distorted general meshes [23, 10, 35, 36, 12], but they are quite expensive and 3- case is still under development [26, 21]. Closely related to the mixed hybrid methods are the finite volume box schemes proposed by Croisille [13, 14]. The proposed method in considerable extent alleviates the drawbacks of the known meth- 3

4 ods without significant increase of the computational cost. The rest of the paper is organized as follows. In Sec. 2 we consider a simple model problem. The discretization is derived in Sec. 3. All necessary formulas for the implementation of the method are provided. Review of the theory for mixed finite element methods is outlined in Sec. 4 and theoretical estimates are proven in Sec. 5. Some numerical results are presented in Sec. 6. Finally we summarize the paper in the conclusions. 2 Model Problem Consider a model boundary value problem: div(k p(x)) = f(x) in, (1a) p(x) = 0 on, (1b) where is a domain in R d, d = 2 or 3, and the boundary is polygonal and convex. The diffusion tensor K is assumed to be a uniformly symmetric positive definite matrix. efining the velocity u as u = K p, we can rewrite (1a) as a first order system for the unknowns u and p u + K p = 0, div(u) = f. (2a) (2b) Note that in some application K can be discontinuous (heat transfer, reservoir simulation), although the normal component of u is continuous. Problems (1) and (2) are equivalent. The weak formulation of problem (1) is: Find (u, p) U P such that u v + K p v = 0 v V, (3a) div(u)q = fq q Q., (3b) 4

5 Note that the test and trial spaces in (3) can be different. Similar methods have been considered in [35, 36] with particular choice U = H(div, ), P = H 1 0 (), V = ( L 2 () )d, Q = L 2 (), where H 1 0() = {p L 2 (), p L 2 (), p = 0 on }, H(div, ) = {v (L 2 ()) d, div(v) L 2 ()}. It is generally difficult to find a stable pair discrete spaces (U h, P h ), U h H(div, ), P h H0 1 () for unstructured meshes. We would like to obtain a discrete method that replaces equation (3b) with an equation similar to the one derived from finite volume methods. Suppose the domain is divided into set of control volumes V i, i = 1,..., n, and assume q is smooth enough. With integration by parts, we can rewrite (3b) as n [ ] u nq q u = V i V i i=1 fq. (4) Choosing Q h to be the space of piecewise constants on the control volumes, Eq. (4) becomes n u h n q = V i i=1 which is a typical equation in finite volume methods. f q, (5) The equation (5) can be considered as an approximation of Eq. (3b) with non-conforming spaces U h and Q h. 3 iscretization We introduce two different meshes that are dual to each other. The meaning of dual will become clear from the following example on which we build the discretization. In particular, the triangular (tetrahedral) mesh, e.g., the elaunay triangulation h, is used 5

6 Figure 1: Voronoi box/elaunay triangles for the discretization of pressure p (the triangles on Fig. 1). The velocity u is approximated on a control volume mesh built from the triangles, for example the Voronoi (PEBI) grid V h. One Voronoi control volume is depicted on Fig. 1 with dotted line. We take the finite element methodology to construct the discretization. The first step is to replace the infinitely dimensional spaces U, P,V, and Q in problem (3) with finite subspaces U h, P h, V h, and Q h. The subindex h is used to emphasize the sequences of spaces for h 0, i.e., when the meshes become more refined, the dimension of the spaces increases. enote with E h the edges of control volumes in V h. We approximate the velocity u with a piecewise constant vector function that have continuous normal components on E h. More details will be given later on the construction of U h. The discrete space P h of approximated pressure is defined on the triangulation h. Every element p P h is a continuous piecewise linear function. We introduce a discrete subspace Q h of L 2 () that consists of piecewise constants on control volumes V V h. Then the discrete problem is defined as follows: 6

7 Find (u h, p h ) U h P h such that u h v h + Note that U h = V h. (x k, y k ) k l j l k ( x, ȳ) i l j i (x i, y i ) (x j, y j ) V V h Figure 2: Triangle V K p h v = 0 v h U h, (6a) u h n q h = fq h q h Q h. (6b) The second step in the finite element discretization is to construct basis for U h. 3.1 Basis of U h We would like to determine the velocity in such a way that the normal components on the edges of the control volumes in any given triangle are continuous. Consider a triangle = i j k as shown in Fig. 2. In each of the quadrilaterals i, j and k the velocity v is approximated with a constant function (six degrees of freedom). We additionally impose that the normal components on l i, l j, l k are continuous (subtract three degrees of freedom). Therefore, there are three degrees of freedom for the velocity space U h in. We choose them to be the integrals of the normal components on l i, l j, l k, i.e., any vector function v U h can be uniquely determined by the numbers v i = v n i, v j = v n j, v k = v n k. l i l j l k 7

8 Note that the values of the velocity in one triangle are not directly connected with the values in the neighboring triangles. Consider vector functions e i, e j, e k defined with the relations e i n i = 1, e i n j = 0, e i n k = 0, l i l j l k e j n i = 0, e j n j = 1, e j n k = 0, l i l j l k e k n i = 0, e k n j = 0, e k n k = 1. l i l j l k (7a) (7b) (7c) It is easy to see that e i, e j, e k are linearly independent and therefore form a basis. So v restricted to the triangle is equal to v = v i e i + v j e j + v k e k. Each of the vectors e i, e j and e k, is defined on the whole triangle, being piecewise constant on each of the quadrilaterals i, j and k. One can compute the basis vectors once the normal vectors n i, n j, n k, and the lengths of the intervals l i, l j, l k are given. For example, e i can be represented as e i = ( ) (e (i) 1 i, e (i) 2 i ), (e (i) 1 j, e (i) 2 j ), (e (i) 1 k, e (i) 2 k ). According to the conditions (7a), in the quadrilateral i we have the identities l i e i n i = 1 and l k e i n k = 0 that lead to the linear system for the unknown components e (i) 1 i n (i) 1 + e (i) 2 i n (i) 2 = 1/ l i, e (i) 1 i n (k) 1 + e (i) 2 i n (k) 2 = 0. Similarly we compute e (i) 1 j and e (i) 1 k. Permuting appropriately the indexes i, j and k in the expressions above we can get the formulas for e j and e k. 8

9 3.2 Assembling The last step is to derive the actual finite volume scheme, in finite element terminology, to assemble the matrix. Consider Eq. (6a) with v h = e i, e j and e k, i.e., u i e i e i + u j e j e i + u k e k e i + K p e i = 0, u i e i e j + u j e j e j + u k e k e j + K p e j = 0, u i e i e k + u j e j e k + u k e k e k + K p e k = 0. Here we have taken into account that the restriction of u h to the triangle is u h = u i e i + u j e j + u k e k. Note that the matrix {e m e n } m,n=i,j,k is symmetric and positive definite. Therefore, we can solve for the fluxes u i, u j and u k and substitute them into Eq. (6b). Let t mn = e m e n, and denote the matrix On the triangle, we have T 1 = t ii t ji t ki t ij t jj t kj t ik t jk t kk. p = p i q i + p j q j + p k q k, where q i, q j and q k are the basis hat functions. enote and m mn = M = We call M the mass matrix. Then u i t ii t ji t ki u j = t ij t jj t kj u k t ik t jk t kk K q m e n m ii m ji m ki m ij m jj m kj m ik m jk m kk 1. m ii m ji m ki m ij m jj m kj m ik m jk m kk p i p j p k = T Mp. 9

10 Having expressed u h through p h, we obtain the final equation for each p i by substituting u h into (6b) and testing with the basis of Q h. Let 1, 2,..., m are the triangles in the support of q i. Then the equation for p i is T i M i p i n q h = fq h. V V V h Here the subscript means T i, M i and p i are defined on triangle i. Remark 1 The proposed method belongs to the class of K methods [19]. One can obtain a K 1 method by first rewriting Eq. (2a) as K 1 u + p = 0, and then following the above procedure. 4 Theoretical Framework In this section we state a theorem which is an easy generalization of the results by Nicolaides [27], Brezzi and Fortin [8] and Thomas and Trujillo [35, 36]. Take an abstract problem Find (u, p) U P a(u, v) + b(v, p) =< g, v > v V, (8a) c(u, q) =< f, q > q Q, (8b) where (U,. U ), (P,. P ), (V,. V ) and (Q,. Q ) are four Hilbert spaces, a(.,.), b(.,.) and c(.,.) are bilinear forms defined respectively on U V, P V and U Q. The right hand sides are defined for g V, f Q, where V and Q are the dual spaces of V and Q correspondingly. 10

11 Consider the discrete problem: Find (u h, p h ) U h P h a h (u h, v h ) + b h (v h, p h ) =< g, v h > h v h V h, (9a) c h (u h, q h ) = < f, q h > h q h Q h, (9b) where (U h,. Uh ), (P h,. Ph ), (V h,. Vh ) and (Q h,. Qh ) are four Hilbert spaces, a h (.,.), b h (.,.) and c h (.,.) are bilinear forms defined respectively on U h V h, P h V h and U h Q h. Suppose that U h U and V h V, i.e., discretization (9) is non-conforming. Let U 0h and V 1h be the spaces defined by U 0h = {u h U h, q h Q h, c h (u h, q h ) = 0}, V 1h = {v h V h, u h U 0h, a h (u h, v h ) = 0}. Assume that there exists three constants A, B and C independent of h such that a h (u h, v h ) A u h Uh v h Vh, b h (v h, p h ) B v h Vh p h Ph. c h (u h, q h ) C u h Uh q h Qh. (10a) (10b) (10c) Then we have the following result: Theorem 1 Assume that the next three Babu ska-brezzi conditions are satisfied: and that a h (u h, v h ) i) inf sup α, (11a) u h U 0h v h V h u h Uh v h Vh ii) iii) b h (v h, p h ) inf sup β, (11b) p h P h v h V 1h v h Vh p h Ph c h (u h, q h ) inf sup γ. (11c) q h Q h u h U h u h Uh q h Qh dim(u h ) + dim(p h ) = dim(v h ) + dim(q h ). 11

12 Then problem (9) has a unique solution (u h, p h ). Moreover, if α, β and γ are independent of h, there exists positive constant C independent of h such that u u h Uh + p p h Ph ( C where inf u v h Vh + inf p p h Ph + M 1h + M 2h + M 3h + M 4h v h V h p h P h a h (u, v h ) + b h (v h, p) < q, v h > M 1h = sup, v h V h v h Vh < g, v h > < g, v h > h M 2h = sup, v h V h v h Vh c h (u, q h ) < f, q h > M 3h = sup, q h q h q h Qh < f, q h > < f, q h > h M 4h = sup. q h Q h q h Qh ), (12) We use the results above for the bilinear forms: a(u, v) = u v, a h (u h, v h ) = u h v h, b(v, p) = K p v, b h (v h, p h ) = K p h v h, c(u, q) = div(u)q, c h (u h, q h ) = u h nq h, V V V where U h U = H(div, ), V h V = (L 2 ()) 2, P h P = H 1 0(), Q h Q = L 2 (). 5 Stability and Error Estimates We start with several auxiliary results. The primary triangular mesh is assumed to be quasiregular as in the analysis of conforming finite element methods. The dual mesh, i.e., the control volumes, are formed by connecting the centers of gravity of the triangles to their edge midpoints. The same error estimates also hold for the three dimensional case when the dual mesh is constructed in a similar way. Some proofs are omitted for simplicity. 12

13 Lemma 1 u h 2 0, [ E ij E h E ij u h n (The is use to indicate the equivalence of the norms.) ] 2, u h U h Proof: The proof follows from the definition of the space U h. The error estimate for the constant interpolant in V h is provided below. Lemma 2 v I c hv 0, Ch v 1,, where I c h v = h [ l i v n i ] [ ] [ ] e i + v n j e j + v n k e k. l j l k Proof: Consider the reference element ˆ and the constant interpolant Îc. We have ( Îcˆv 0, ˆ C( ˆ) ) ˆv n i e i + ˆv n l i j l j e j + ˆv n k e k l k C 1 ( ˆ)( ˆv 0,li + ˆv 0,lj + ˆv 0,lk ) C 2 ˆv 1, ˆ. Therefore, ˆv Îcˆv 0, ˆ C 2 ˆv 1, ˆ. By the Bramble-Hilbert lemma argument we get ˆv Îcˆv 0, ˆ C 2 ˆv 1, ˆ. Transformation to the original element produces the desired result. In order to use Theorem 1 one has to specify the spaces and norms mentioned in it. We define the norms as p h Ph = p h 1,, q h Qh = (q i q j ) 2 xi ω u h Uh = u h Vh = u h 0,, j Π(i) 1/2, 13

14 where ω consists of the nodes of the triangles in the triangulation h and Π(i) is the set of neighbors of x i. Recall a well known result for the equivalence of the norm on P h with H0 1 -seminorm [9]: Lemma 3 p h Ph p h 1, p h P h. The following theorem is the error estimate for the method proposed in this paper. Theorem 2 Let (u, p) be the solution of problem (3) and u (H 1 ()) 2, p H 1 (). Suppose that the triangular grid is quasiregular and the control volumes are constructed by connecting the center of gravity of the triangles to their edge midpoints. Then there exist an unique solution (u h, p h ) of (6) and a positive constant C such that p p h 1, + u u h 0, Ch( p 2, + u 1, ). Proof: It is easy to verify (10a) and (11a) for the bilinear form a h (.,.). We check the conditions (10b) and (11b) for the bilinear form b h (.,.) as follows. Since K is a uniformly positive definite matrix, there exists positive constants k 1 and k 2 such that So, b h (v h, p h ) = ( v h K p h k 1 ξ t ξ ξ t Kξ k 2 ξ t ξ, ξ R d. Kv h v h ) 1/2 ( Choosing v h (p h ) = p h. Then ( b h (v h (p h ), p h ) = v h (p h ) K p h = k 1 v h (p h ) 0, p h 1, C v h (p h ) 0, p h 1,. K p h p h ) 1/2 k 2 v h 0, p h 1,. ) 1/2 ( ) 1/2 Kv h (p h ) v h (p h ) K p h p h Hence, in order to verify that (11b) is true, it suffices to show that p h V 1h, p h P h, i.e., we only need to show a h (u h, p h ) = h u h p h = 0, u h U 0h, p h P h. (13) 14

15 Consider a triangle as shown in Fig. 2. enote the normal vectors for l i, l j, l k as ñ i, ñ j, ñ k with the direction i j k i. Note that ñ i, ñ j, ñ k have length l i, l j, l k respectively, different from those normal vectors defined in Sec Let s be the area of triangle. Take N i, N j and N k to be the outwards normal vectors of triangle. The usual convention applies, i.e., N i is the normal vector of the edge (x j, y j ) (x k, y k ), and the norm of N i is equal to the length of the corresponding edge. Since the edge midpoints and center of gravity are used to form the control volumes, one can show i = j = k = s 3, ñ i ñ j + ñ k = 0, N i = 2(ñ i ñ k ), N j = 2(ñ j ñ i ), N k = 2(ñ k ñ j ). Since p h is a linear function on, we have p h = 1 2s (p in i + p j N j + p k N k ). Therefore, u h p h = u h p h i + j + k = 1 ( ) uh i + u h j + u h k (pi (ñ i ñ k ) + p j (ñ j ñ i ) + p k (ñ k ñ j )), 3 where the coefficient of ( 1p 3 i) is equal to ( ) uh i + u h j + u h k (ñi ñ k ) = u h i (ñ i ñ k ) + u h j (ñ i ñ k ) + u h k (ñ i ñ k ) = u h i (ñ i ñ k ) + u h j (2ñ i ñ j ) + u h k (ñ j 2ñ k ) = u h i (ñ i ñ k ) + 2u h j ñ i u h j ñ j + u h k ñ j 2u h k ñ k = u h i (ñ i ñ k ) + 2u h i ñ i u h j ñ j + u h j ñ j 2u h i ñ k = 3u h i (ñ i ñ k ) In the above equalities, we use the fact that the normal component of u h is continuous across the edges l i, l j, l k. So, u h p h = ( p i u h i (ñ i ñ k ) + p j u h j (ñ j ñ i ) + p k u h k (ñ k ñ j ) ). 15

16 Substitute it into Eq. (13) and sum all the coefficients of p i. Then, a h (u h, p h ) = u h n p V = 0, V V V h where p V denotes the value of p h at the center of control volume V and the last equality follows from the fact u h U 0h. Now we check the conditions (10c) and (11c). The condition (10c) can be easily verified. Given q h Q h, construct u(q h ) such that u(q h ) n Eij E ij = q i q j. Then c h (u(q h ), q h ) = u(q h ) n q h = V V h = V E ij E h u(q h ) n Eij E ij (q i q j ) E ij E h [ ] 2 1/2 u(q h ) n 1/2 (q i q j ) 2 E ij E ij E h C u(q h ) 0, q h 1,. Therefore, c h (u, q h ) sup c h(u(q h ), q h ) C q h 1,. u U h u 0, u(q h ) 0, To finish the proof of the theorem we note that M 1h, M 2h, M 3h and M 4h are equal to zero for the corresponding bilinear forms and Lemma 2 provides a bound for the interpolation of u in (12). 6 Numerical Results In this section we present numerical experiments with three model problems on a domain - a pentagon with a square hole inside. The coordinates of the pentagon vertexes are (( 1, 0), (1, 0), (1.6, 2), (0, 4), ( 1.6, 2)) and the coordinates of the square hole are (( 0.7, 0.9), (0.7, 0.9), (0.7, 2.3), ( 0.7, 2.3)). The domain is shown on Fig. 3 with 66 mesh points and 92 triangles. We choose such a domain to illustrate the flexibility of the finite volume method. The elaunay triangulations are generated using the software product triangle developed by J. R. Shewchuk [34]. Triangle is a elaunay triangulator and it provides the control 16

17 Figure 3: omain and mesh points on the maximum area of the triangles. We used this option to generate five elaunay triangulations. The maximum area of the triangles decreases four times on every successive level. The number of nodes increase roughly by four. The control volumes are created by connecting the center of gravity of each triangle to the center of its edges. Problem 1 (Laplacian) Take K(x) = I and the exact solution p(x, y) = sin(πx) sin(πy). Problem 2 (Smooth full tensor (see Shashkov [33])) Take K(x, y) = R (x, y) R T, where [ cos(φ) sin(φ) R = sin(φ) cos(φ) ] [ ] d1 0, (x, y) = 0 d 2 17

18 and φ = π 4, d 1 = 1 + 2x 2 + y 2 + y 5, d 2 = 1 + x 2 + 2y 2 + x 3, and the exact solution is p(x, y) = sin(πx) sin(πy). Problem 3 (iscontinuous full tensor) Let [ ] K(x, y) = [ ] for x 0, for x > 0. The exact solution is given as p(x, y) = { (1 + x)y, for x 0, (1 + x)y + x(1 + y), for x > 0. We compute the L 2 error of p h and u h, the H 1 error of p h, and the H(div) h error for u h. In particular, the H(div) h error takes the form u u h H(div)h = ( V V h V (u h n u n) dl) 1/2 = ( V V h V (u h n + K p n) 2 dl) 1/2. The numerical results reported in Tab. 1-3 confirm the first order error estimates for p h and u h proven in Theorem 2. We also observe that there is second order convergence of the pressure in L 2 -norm and half an order convergence for the discrete divergence of the velocity. Remark 2 Similar results are obtained for all the three test problems with the K 1 method discussed in Remark 1. 18

19 Table 1: iscrete norms of the error for Problem 1 N p p h 0, p p h 1, u u h 0, u u h H(div),h Order Table 2: iscrete norms of the error for Problem 2 N p p h 0, p p h 1, u u h 0, u u h H(div),h Order Table 3: iscrete norms of the error for Problem 3 N p p h 0, p p h 1, u u h 0, u u h H(div),h Order

20 7 Conclusion We have developed a locally conservative and flux-continuous K method for the elliptic diffusion equation with tensor diffusion tensor. The method is well defined on very general meshes and is directly applicable to 3- problems. Stability and error estimate are also established when the control volumes are formed by connecting the center of gravity of the triangles to their edge midpoints. Numerical experiments confirm our theoretical results. Similar numerical results were obtained for its corresponding K 1 method. The same error estimates can be shown for the three dimensional case, and will be reported elsewhere. There are still several theoretical and practical issues left unresolved like the convergence of the pressure in L 2 -norm and convergence of the velocity in the discrete H(div)-norm. References [1] I. Aavatsmark, T. Barkve, O. Bœ, and T. Mannseth. iscretization on unstructured grids for inhomogeneous, anisotropic media. Part I: erivation of the methods. SIAM J. Sci. Stat. Comput., 19: , [2] L. Angermann. An introduction to finite volume methods for linear elliptic equations of second order. Technical Report No. 163, Friedrich-Alexander University of Erlangen- Nuremberg, Erlangen, [3] T. Arbogast, C. N. awson, P. T. Keenan, M. F. Wheeler, and I. Yotov. Enhanced cell-centered finite differences for elliptic operators on general geometry. SIAM J.Sci. Comput., 19(2): , [4] T. Arbogast, M. F. Wheeler, and I. Yotov. Mixed finite elements for elliptic problems with tensor coefficients as cell centered finite differences. SIAM J.Numer. Anal., 34(2): ,

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