Arbogast, T., Wheeler, M.F. and Yotov, I.: Mixed finite elements for elliptic problems with tensor coefficients as cellcentered finite differences, SIAM J. Numer. Anal., 34 (1997).  Home/Search   Document Details and Download   Summary   Related Articles   Check

....the solution can be approximated accurately. Cell Centered Finite Differences on Rectangular Meshes An expanded mixed finite element approximation for second order elliptic problems (such as flow or dispersion diffusion problems) containing a tensor coefficient was developed and investigated [13, 42]. The method is expanded in the sense that three variables are explicitly approximated, the scalar unknown (pressure or concentration) its gradient, and its flux (velocity) Optimal order error estimates and superconvergence of the scalar variable were shown. The scheme is suitable for the case ....

....permeability or small diffusion) since it does not need to be inverted. In the case of the lowest order Raviart Thomas elements [132] on rectangular parallelepipeds, this expanded mixed method may be simplified to a cell centered finite difference method by incorporating certain quadrature rules [13]. For a general tensor coefficient, the sparsity pattern for the scalar unknown is a nine point stencil in two dimensions, and 19 points in three dimensions. Such a compact stencil is well suited to parallel computation. Moreover, it requires the solution of a positive definite linear system. ....

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T. Arbogast, M. F. Wheeler, and I. Yotov, Mixed finite elements for elliptic problems with tensor coefficients as cell-centered finite differences, (submitted).

....two space dimensions) or Voronoi meshes; let us also mention [8] and [9] where more general meshes are treated, with, however, a somewhat technical geometrical condition. In the pure diffusion case, the cell centered finite volume method has also been analyzed with finite element tools: 1] 4] [2] or Petrov Galerkin tools [10] The convergence analysis has also been performed in some cases of nonlinear convection diffusion problems; see [14] with a combined finite element finite volume method, 13] and [12] with a pure finite volume scheme. Since the approximate solution constructed with a ....

Arbogast, T., M.F. Wheeler and I. Yotov(1997), Mixed finite elements for elliptic problems with tensor coefficients as cell-centered finite differences, SIAM J. Numer. Anal. 34, 2, 828-852.

....Figure 2) We also retain the integration of k Gamma1 as in Eq. 5) which will generalize the usual harmonic averaging of k in a simple way. An expanded mixed method, reducible to a finite difference scheme by low order integration, has been formulated, analyzed, and tested by Arbogast et al. [3, 4]. The method is expanded in the sense that it introduces an additional variable corresponding to rp, subsequently eliminating it under some circumstances. A key to the method is the assumption that there is a global C 2 mapping from the irregular grid to a regular grid, which is not the case for ....

....has been eliminated. Note that we did not have to require that p be piecewise constant in this derivation, though we will generally think of the numerical approximation of p in this way. The step just completed is the elimination of the analogues of the Lagrange multipliers of Arbogast et al. [3, 4], mentioned in the discussion of other methods for irregular grids in Section 1. The ability to carry out this step is a special property of the control volume mixed method, as opposed to the standard framework in which the vector shape and test functions are the same. In the latter case, the ....

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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., to appear.

....Russell et al. 27] have proposed a conservative control volume mixed method that produces a nonsymmetric matrix for a symmetric differential problem. Thomas has developed mixed finite volume methods [34, 35] that also generates a nonsymmetric matrix. Recently Arbogast, Wheeler and Yotov [2] have generalized the results in [38] for diffusion problems with tensor coefficients and have derived and analyzed new cell centered finite volume difference schemes. We still do not know when or even whether these methods are monotone. The research in this direction has just begun [21] Finite ....

Todd Arbogast, Mary F. Wheeler, and Ivan Yotov. Mixed finite elements for elliptic problems with tensor coefficients as cell--centered finite differences. SIAM J.Numer. Anal., to appear.

....the case of triangular or Voronoi meshes; let us also mention [9] and [7] where more general meshes are treated, with, however, a somewhat technical geometrical condition. In the pure diffusion case, the cell centered finite volume method has also been analyzed with finite element tools: 1] 4] [2]. The convergence analysis has also been performed in some cases of nonlinear convection diffusion problems; see [19] with a combined finite element finite volume method, 14] and [17] with a pure finite volume scheme. A first order estimate for triangular meshes was obtained in [26] for a ....

Arbogast, T., M.F. Wheeler and I. Yotov(1997), Mixed finite elements for elliptic problems with tensor coefficients as cell-centered finite differences, SIAM J. Numer. Anal. 34, 2, 828-852.

....with two remarks. First, it has been recently shown that the linear system arising from expanded mixed methods based on some lower order mixed finite elements for linear second order differential problems can be thought of as a linear system generated by cell centered finite difference methods [1] where superconvergence results were also obtained. Second, an expanded mixed formulation has been also used for elasticity problems (see [12] and the bibliographies therein) and for second order elliptic problems in [17] However, the detailed analysis of the expanded mixed methods is carried ....

T. Arbogast, M. Wheeler, and I. Yotov, Mixed finite elements for elliptic problems with tensor coefficients as finite differences, SIAM J. Numer. Anal. 34 (1997), to appear.

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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):828--852, 1997.

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T. Arbogast, M.F. Wheeler and I. Yotov, Mixed finite elements for elliptic problems with tensor coeffcients as cell-centered finite differences, SIAM J. Numer. Anal. 34(2) (1997) 828--852.

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ARBOGAST, T., WHEELER, M. F., and YOTOV, I., Mixed Finite Elements for Elliptic Problems with Tensor Coefficients as Cell-Centered Finite Differences, SIAM Journal on Numerical Analysis, Vol. 34, pp. 828--852, 1997.

....implicit two phase model Here we describe the multiblock mortar formulation of the implicit two phase model as given by equations (1) 2) The other models are discretized similarly. We omit the details. We employ a variant of the mixed finite element method, the expanded mixed method following [6]. It has been developed for accurate and efficient treatment of irregular domains (see [6,4] for single block and [55] for multiblock domains) In the context of multiphase flow this method allows for proper treatment of the degeneracies in the diffusion term (see remark 3.1) See also [42] For ....

....two phase model as given by equations (1) 2) The other models are discretized similarly. We omit the details. We employ a variant of the mixed finite element method, the expanded mixed method following [6] It has been developed for accurate and efficient treatment of irregular domains (see [6,4] for single block and [55] for multiblock domains) In the context of multiphase flow this method allows for proper treatment of the degeneracies in the diffusion term (see remark 3.1) See also [42] For m w, o we define = #Pm . Then = k mK m# mG#D . The implicit in time ....

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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 (1997) 828--852.

....such that (8) V. nq, w) V .q, w) w e W, 9) IIq nqll 5 Cllqllk, 10) IIv (q nq)ll 5 cllv qllk. We wi use the estimate [5] 11) IlHu n unIITM IIH n finllTM 5 Ck = where, again, we let k = h or H. We wi also make use of the foowing lemma proven in Arbogast, Wheeler and Yotov [1]. LMMX 2.1. For the lowest order RTN spaces on rectangles, for any q = q,qY) HI( and E Oq (12) II(nq)110, Ilex x 0, y OqY (13) II (IIq)Yllo,E II llo,E. In the following arguments, C will represent a generic constant independent of H, h and At. We will use the standard inequality, ....

.... satisfying (18) 19) 20) dtP, w) V U , w) fn, W) VW WH, 0 , V)TM: P, V v) Vv V , U , V)TM (K(PH(P) Ot, V)T, VV VH, and we take P = PH(t , This scheme is based on an expansion of the standard mixed finite element method that was formulated for linear elliptic problems in [1]. We define PH(P) from the values of Pii for i = 1, x and j = 1, y as follows. For points (x,y) such that xi x Xi l,i 1, x and Yi Y Yi I,J 1, y , we take PH(p) x, y) to be the bilinear interpolant, PH(p) x,y) X X i Yj I Y (Pij( i 1 ) q Pi lj( ....

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T. ARBOGAST, M. F. WHEELER, AND I. YOTOV, Mixed finite elements for elliptic problems with tensor coefficients as cell-centeredfinite differences, Dept. Comp. Appl. Math. TR95-06, Rice University, Houston, TX 77251, Mar. 1995.

.... and superconvergence estimates can be found, for example, in [19, 8, 16, 23, 10, 11] Since the standard MM yields a linear system that represents a saddle point problem, much current research on the MM involves how to efficiently solve the system of equations that arises, see for example, [13, 5, 6, 12, 21, 17, 1]. Perhaps the earliest successful technique was the hybrid form of the mixed method (HM) 3] This turns the saddle point problem into a semi definite problem, but at the expense of greatly increasing the number of unknowns. This research was supported in part by the Department of Energy, the ....

....and prove superconvergence for the tential and velocity approximations generated by this method. The resulting matrix problem is definite and generally much easier to solve than the problem which arises using the standard mixed method without quadrature. Recently, a variation of the mixed method [1, 2] has been developed which has advantages over the standard approach for tensor and positive semi definite coefficients K. In this method, an auxiliary variable is introduced to avoid inverting K, allowing K to be nonnegative the standard mixed method assumes K is strictly positive. This ....

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T. Arbogast, M. F. Wheeler, and I. Yotov, Mixed Finite Elements for Elliptic Problems with Tensor Coejcients as Finite Differences. Technical Report TR94-02, Department of Computational and Applied Mathematics, Rice University, 1994.

....cell centered finite difference formulation (CCFD) to (2.1) to get Ot (V. Vc) uc Next, we discretize Eq. 2.2) in order to define (V. Vo) ij. Note that, under certain circumstances, CCFD is equivalent, up to quadrature error, to the expanded mixed finite element method using RT0 spaces [34, 5, 3]. In particular, discrete form of (2.2) in i th direction reads, with gravity terms omitted for simplicity (2.4) U W2 J = P ) where l 2, k, l 2, k on he edge i 1 2,j, k between cells i,j, k d i 1,jk, w and w and by are obtained by harmonic averaging of permeabilities hor hor ....

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):828-852, 1997.

....(i.e. O(N Gamma2 ) at Gaussian points and O(N Gamma2 ) global L 2 estimate for both p and a 1=2 5 p are obtained by a local postprocessing. Modifications in the theoretical analyses can be extended to treat cell centered finite differences, RT 0 with numerical quadrature [29, 24, 35, 3, 2]. Numerical results in this case which support our theory are presented and show that the anisotropic mesh gives more accurate results than the standard uniform mesh. The organization of this paper is as follows. In x2, a general MFEM with exact quadrature is presented for (1.1) For completeness, ....

.... Gamma2 : 4.44) Theorem 4.8. Under the assumptions of Lemma 4.5, we have (jjjp x Gamma p h x jjj 1 jjjp y Gamma p h y jjj 2 ) jjjp Gamma p h jjj 0 CN Gamma2 : 4.45) 5. Numerical results. In this section we present some numerical tests on the cell centered finite difference method [29, 24, 35, 3, 2], since the cell centered finite difference scheme is equivalent to the rectangular RT 0 MFEM with special numerical quadrature formulae [29] Similar theoretical analysis can be obtained by proper modifications of the proofs in [24, 35, 3, 2] To check our theoretical analysis, we tested an ....

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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 (1997), pp. 828--852.

.... (which preserves the normal component of vectors across boundaries) Since the standard implementation of the mixed method yields a linear system that represents a saddle point problem (as in x3.1 below) much current research involves how to efficiently solve such systems (see, e.g. [32, 22, 21, 33, 14, 29, 13, 2]) This is a particular problem when the domain is irregular. The difficulty of the solution process can be eased by reformulating or further approximating the mixed method so that it yields a positive definite linear system. As Arnold and Brezzi [3] pointed out, this can be done directly by using ....

....difference methods have been the standard approach for many years [30] The relationship between the mixed method and cell centered finite differences on rectangular grids was first established in [32] under the assumption that K in (1. 1) is a scalar or a diagonal matrix, and later in general in [2] for a variant of the mixed method, the expanded mixed method [37, 25, 10, 2] The primary mixed finite element methods on general geometry 3 restrictive assumption is that the mesh is rectangular. If one uses the RT 0 space and applies appropriate quadrature rules, the velocity unknowns can ....

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T. Arbogast, M. F. Wheeler, and I. Yotov, Mixed finite elements for elliptic problems with tensor coefficients as cell-centered finite differences, to appear.

....Science Foundation Center for Research on Parallel Computation, and Batalle Pacific Northwest Laboratory. y University of Texas at Austin and Rice University 2 MARY F. WHEELER AND IVAN YOTOV mixed finite element methods for single phase flow. First, the expanded mixed finite element method [49, 38, 16, 4] is formulated to treat general geometry problems and full tensor permeability or dispersion. In addition special quadratures are introduced to obtain a cell centered finite difference procedure for treating logically rectangular grids. Extensions to discontinuous coefficients, multiblock domains, ....

....Let h Delta; Deltai S denote the L 2 ( S) inner product or duality pairing. Define H(div; S) fv 2 (L 2 (S) d : r Delta v 2 L 2 (S)g; with the norm kvk H(div;S) Z S i jvj 2 jr Delta vj 2 j dx ) 1=2 : MIXED METHODS FOR FLOW AND TRANSPORT IN POROUS MEDIA 3 Following [4, 1] we introduce the adjusted pressure gradient u = GammaM Gamma1 rp; where M is some symmetric positive definite tensor related to the geometry of Omega Gamma Then u = KM u; and we have the following expanded mixed formulation. Mu; v) MKM u; v) v 2 (L 2( Omega Gamma9 d ; 2.5) ....

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T. Arbogast, M. F. Wheeler, and I. Yotov, Mixed finite elements for elliptic problems with tensor coefficients as cell-centered finite differences. To appear in SIAM J. Numer. Anal. MIXED METHODS FOR FLOW AND TRANSPORT IN POROUS MEDIA 15

.... On rectangular elements, if K in (1) is a scalar or a diagonal matrix, one finds that numerical quadrature reduces the mixed method to a five point finite difference method for pressure [10] The new AWYM (Arbogast Wheeler Yotov method) extends these results to general coefficient tensors [2, 3, 4]. On triangular elements, however, the linear system arising from the MFEM is sparse but indefinite, making it expensive to solve. We have developed a cell centered stencil method (CCSM) which reduces the MFEM on triangular elements to a ten point finite difference type stencil. This provides a ....

....approximating spaces are used, the pressure and velocity are globally first order accurate. Superconvergence of pressure to second order is observed at the center of mass of each cell, and for rectangular meshes, superconvergence of velocities to second order is observed at certain Gauss points [2, 3]. We note that if S were a diagonal matrix, then Aawym would be sparse. Recently we developed the cell centered stencil method (CCSM) for triangular meshes [1] which reduces S to a diagonal matrix by numerical integration of the left side of (8) In two space dimensions, the resulting matrix has ....

Arbogast, T., Wheeler, M.F., and Yotov, I. (in preparation) "Mixed finite elements for elliptic problems with tensor coefficients as finite differences".

.... and superconvergence estimates can be found, for example, in [19, 8, 16, 23, 10, 11] Since the standard MM yields a linear system that represents a saddle point problem, much current research on the MM involves how to efficiently solve the system of equations that arises, see for example, [13, 5, 6, 12, 21, 17, 1]. Perhaps the earliest successful technique was the hybrid form of the mixed method (HM) 3] This turns the saddle point problem into a semi definite problem, but at the expense of greatly increasing the number of unknowns. This research was supported in part by the Department of Energy, the ....

....and prove superconvergence for the potential and velocity approximations generated by this method. The resulting matrix problem is definite and generally much easier to solve than the problem which arises using the standard mixed method without quadrature. Recently, a variation of the mixed method [1, 2] has been developed which has advantages over the standard approach for tensor and positive semi definite coefficients K. In this method, an auxiliary variable is introduced to avoid inverting K, allowing K to be nonnegative the standard mixed method assumes K is strictly positive. This ....

[Article contains additional citation context not shown here]

T. Arbogast, M. F. Wheeler, and I. Yotov, Mixed Finite Elements for Elliptic Problems with Tensor Coefficients as Finite Differences. Technical Report TR94-02, Department of Computational and Applied Mathematics, Rice University, 1994.

....geometrically general domain, we need to develop a new scheme. We will not sacrifice the ease of implementation, the accuracy, or the local mass conservation property of the approximation. We present in the next section the expanded mixed finite element method that is the basis of our scheme [4, 5, 6]. Extensions to the expanded hybrid formulation are also discussed. The hybrid formulation involves introducing Lagrange multipliers on the boundaries of elements or subdomains on which the components of the tensor are discontinuous. This is an important modification to obtain higher order ....

....differs from [8] in that we do not require the mesh to be orthogonal. Omega x F Gamma Gamma Gamma Gamma Omega x Fig. 1. The computational domain Omega and the physical domain Omega Gamma We derive our finite difference procedure [4, 6] in Section 3, and summarize our convergence results [4, 5] in Section 4. Computational experiments are given in Section 5, including experimental convergence results for tensor coefficient problems, an example of a parallel domain decomposition substructuring algorithm for solving the expanded hybrid formulation, and a tracer calculation on a general ....

[Article contains additional citation context not shown here]

Arbogast, T., Wheeler, M.F., and Yotov, I.: "Mixed finite elements for elliptic problems with tensor coefficients as cell-centered finite differences" (in preparation) .

....accuracy, or the local mass conservation property of the approximation. In the notation of the flow problem, we present in the next section some of the necessary background of the expanded mixed finite element method that is the basis of our scheme [4] We derive our finite difference procedure [3] from it in Section 3. We summarize the convergence results [4] in Section 4. Transport is discussed briefly in Section 5, and computational results are given in Section 6. Our main requirement is that there be a smooth mapping F of a rectangular, computational domain Omega onto the aquifer ....

....= jdet(DF )j. There are grid generation codes available for creating F and its Jacobian matrix. Omega x F Gamma Gamma Gamma Gamma Omega x Fig. 1. The computational domain Omega and the physical domain Omega Gamma 2. THEEXPANDED MIXED FINITE ELEMENTMETHODON GENERAL GEOMETRY Following [3] and [4] we introduce an unknown u such that (2.1) M u = Gammarp; u = KM u; where M = J(DF Gamma1 ) T DF Gamma1 . Note that M is a symmetric, positive definite matrix. It is introduced to simplify the computations significantly, after mapping to the rectangular grid on Omega Gamma ....

[Article contains additional citation context not shown here]

Arbogast, T., Wheeler, M.F., and Yotov, I. (in preparation) "Mixed finite elements for elliptic problems with tensor coefficients as finite differences".

....TX 77005 1892, and TICAM, The University of Texas at Austin, Austin, TX 78712; yotov ticam.utexas.edu. 2 ARBOGAST, COWSAR, WHEELER, AND YOTOV A number of papers deal with the analysis and the implementation of the mixed methods applied to the above problem on conforming grids (see, e.g. [25, 23, 22, 7, 5, 6, 9, 12, 21, 26, 13, 15, 2, 1] and [24, 8] Mixed methods on nested locally refined grids are considered in [14, 16] These works apply the notion of slave or worker nodes to force continuity of fluxes across the interfaces. The results rely heavily on the fact that the grids are nested and cannot be extended to ....

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 (1997). To appear.

....piecewise linear finite element Galerkin method, and they used a regularization of the problem to obtain their results. In the petroleum industry, equations similar to (1. 1) see Section 2) are most often discretized by using the cell centered finite difference method [26] As shown in [30] 32] [6], this scheme is actually the lowest order Raviart Thomas mixed finite element method on rectangles [27] combined with special quadrature rules. The mixed method for the nondegenerate problem has been well studied (see, e.g. 27] 14] 21] however, it appears that no convergence theory has ....

....7. Some superconvergence results. In this section, we present some superconvergence results for the nondegenerate case. Such results are known for linear elliptic problems under the hypotheses that ff is diagonal and the grid is rectangular [22] 17] 16] or merely that the grid is rectangular [6]. We need to assume that fi = 0, and that there is a constant C 5 = C 5 (u) such that (A9) jr P h uj C 5 (u) for x 2 E; and E an element of the mesh: This holds, for instance, if W h consists of piecewise constants defined over the given mesh, e.g. the lowest order Raviart Thomas spaces, ....

T. Arbogast, M. F. Wheeler, and I. Yotov, Mixed finite elements for elliptic problems with tensor coefficients as finite differences, to appear.

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T. Arbogast, M. F. Wheeler and I. Yotov: ÒMixed finite elements for elliptic problems with tensor coefficients as cell-centered finite differences,Ó SIAM J. Numer. Anal., 34,

....convex or Omega is smooth enough (see [21, 22, 19] Strictly speaking, this simplification excludes point or line sources and discontinuous K. A number of papers deal with the analysis and the implementation of the mixed methods applied to the above problem on conforming grids (see, e.g. [28, 26, 25, 8, 6, 7, 10, 14, 24, 29, 15, 17, 2, 1] and [27, 9] Mixed methods on nested locally refined grids are considered in [16, 18] These works apply the notion of slave or worker nodes to force continuity of fluxes across the interfaces. The results rely heavily on the fact that the grids are nested and cannot be extended to ....

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 (1997), pp. 828--852.

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Arbogast, T., Wheeler, M.F. and Yotov, I.: Mixed finite elements for elliptic problems with tensor coefficients as cellcentered finite differences, SIAM J. Numer. Anal., 34 (1997).

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