Arbogast, T., Dawson, C.N., Keenan, P.T., Wheeler, M.F. and Yotov, I.: Enhanced cell-centered finite differences for elliptic equations on general geometry, to appear in SIAM J. Sci. Comp., 1997.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....consistent with our face normal component representation. More accurate approximations for the dot product integral have been derived in [10] using an interpolation scheme that is exact for both constant vector functions and a class of vector functions obtained via the so called Piola transform [1,3]. For these functions the interpolated integrand can be integrated exactly using symbolic manipulations [10] The resulting expression for dot product integral is more accurate then the one described here, but it has the same order of convergence. However, the interpolated dot product method ....

T. Arbogast, C.N. Dawson, P.T. Keenan, M.F. Wheeler and I. Yotov, Enhanced cell-centered finite differences for elliptic equations on general geometry, SIAM J. Sci. Comput. 18 (1997) 1--32.

....used by Rose can be derived from the diffusion equation together with the integral identitythatwe use. Rose presented a proof that his hexahedral mesh method converges with second order accuracy, but he provided computational results only for a 1;D version of his method. Arbogast, et al. [6] have recently developed a cell centered expanded mixed finite element method for solving the tensor diffusion equation on general meshes (including hexahedral meshes. Their method has only cell center intensity unknowns if both the mesh and the diffusion tensor are smooth, but additional ....

....added complexity of the basis function support operators method. We feel that our local support operators method for general hexahedral meshes is much simpler 8 than hybrid mixed finite element methods precisely because the vector basis functions for hexahedral meshes are extremely complicated [6]. More importantly, our local supportoperators method converges on non smooth hexahedral meshes, but wehave not been able to identify anyhybrid mixed finite elementmethodsthathave been shown to converge on such meshes. To summarize, the following combination of characteristics appear to be unique ....

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Todd Arbogast, ClintN.Dawson, Philip T. Keenan, Mary F. Wheeler, and Ivan Yotov, "Enhanced Cell-Centered Finite Differences for Elliptic Equations on General Geometry," SIAM J. Sci. Comput., 18, 1 (1997). 57

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T. Arbogast, C.N. Dawson, P.T. Keenan, M.F. Wheeler and I. Yotov, Enhanced cell-centered finite differences for elliptic equations on general geometry, SIAM J. Sci. Comput. 19(2) (1998) 404--425.

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ARBOGAST, T., DAWSON, C. N., KEENAN, P. T., WHEELER, M. F., and YOTOV, I., Enhanced Cell-Centered Finite Differences for Elliptic Equations on General Geometry, SIAM Journal on Scientific and Statistical Computing, Vol. 19, pp. 404--425, 1997.

....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 ....

....permeability tensor K, for accurate approximation of the mixed method on each subdomain by cell centered finite differences for P o and N O . This is achieved by approximating the vector integrals in (13) and (14) by a trapezoidal quadrature rule and eliminating U h,m and U h,m from the system [6,4]. Remark 3.2. The usual piecewise constant Lagrange multiplier space for RT 0 leads to only O(1) approximation on the interfaces in the case of non matching grids. With the above choice for mortar space, optimal convergence and, in some cases, superconvergence is recovered for both pressure and ....

T. Arbogast, C.N. Dawson, P.T. Keenan, M.F. Wheeler and I. Yotov, Enhanced cell-centered finite differences for elliptic equations on general geometry, SIAM J. Sci. Comput. 19 (1998) 404--425.

....four projection operators, the proof is st relatively simple extension of that presented by Douglas and Roberts [15] Chen [11] 12] also analyzed st similar expanded mixed method. We present briefly the proof here for completeness and for later analysis of the finite difference scheme (see also [1], where st somewhat more general expanded mixed finite element method is studied) From (1.3) with (1.3b) and (1.3d) extended to P F) and (2.2) we get the error (3.243) V. IIu un) w) 0, w C (3.24b) Ifi ah,v) 7)hp ph,X7.v) Qhh hh,v.v)r , vcVh, 3.24c) u ua, K(fi aa) ff) e ....

....or [24] We show in the next section that our scheme has global convergence properties. Moreover, it is symmetric and locally conservative, and it has a compact 9 or 19 point stencil and connections to mixed finite element methods. Moreover, it can be extended easily to nonrectangular grids (see [1], 3] and [2] 5. An error analysis of the finite difference method. For either quadrature rule Q, let x denote the characteristic function of the set and extend the definition of the discrete inner products to (w,w)q,s = w,ws)q For w C W N C( v C N (C( d, and h implicitly fixed, let ....

T. ARBOGAST, C. N. DAWSON, P. T. KEENAN, M. F. WHEELER, AND I. YOTOV, Enhanced cell-centered finite differences for elliptic equations on general geometry, SIAM J. Sci. Comput., to appear.

....domain fla covering fl, which is composed of rectangular ijk grid cells. For simplicity we assume fla = fl, In general, fla doesn t have to be a parallelepiped and any curved boundary can be approximated by keyed out cells. Moreover, the grid must only be locally logically rectangular [4, 3]. We apply 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 ....

....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, C. N. Dawson, P. T. Keenan, M. F. Wheeler, and I. Yotov. Enhanced cell- centered finite differences for elliptic equations on general geometry. SIAM J. Sci. Comp., 19(2):404-425, 1998.

....method and implicitly approximate diffusive terms using a mixed finite element method. Recently, Dawson and Aizinger [19] extended this analysis by applying the Discontinuous Galerkin method developed by Cockburn and Shu [11] and the Enhanced Mixed finite element method developed by Arbogast et al. [1] to the transport equation. They analyzed the standard transport equation utilizing higher order approximating spaces, a positive semi definite diffusion coefficient, and physically realistic boundary conditions. Recently, Arbogast and Wheeler developed the Characteristic Mixed method in [2] ....

T. ARBOGAST, C.N. DAWSON, P.T. KEENAN, M.F. WHEELER, and I. YOTOV. Enhanced cell-centered finite differences for elliptic equations on general geometry. SIAM J. Sci. Comput., 19(2):404--425, 1998.

....(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 ....

[Article contains additional citation context not shown here]

T. Arbogast, C.N. Dawson, P.T. Keenan, M.F. Wheeler and I. Yotov, Enhanced cellcentered finite differences for elliptic equations on general geometry, SIAM J. Sci. Comput. 19 (1998), pp. 404--425.

....four projection operators, the proof is a relatively simple extension of that presented by Douglas and Roberts [15] Chen [11] 12] also analyzed a similar expanded mixed method. We present briefly the proof here for completeness and for later analysis of the finite difference scheme (see also [1], where a somewhat more general expanded mixed finite element method is studied) From (1.3) with (1.3b) and (1.3d) extended to Gamma F ) and (2.2) we get the error equations (r Delta ( Piu Gamma u h ) w) 0; w 2 W h ; 3.24a) Pi u Gamma u h ; v) P h p Gamma p h ; r Delta v) ....

....or [24] We show in the next section that our scheme has global convergence properties. Moreover, it is symmetric and locally conservative, and it has a compact 9 or 19 point stencil and connections to mixed finite element methods. Moreover, it can be extended easily to nonrectangular grids (see [1], 3] and [2] 5. An error analysis of the finite difference method. For either quadrature rule Q, let S denote the characteristic function of the set S and extend the definition of the discrete inner products to (w; w)Q;S = w; w S ) Q For w 2 W C 0 ( Omega Gamma1 v 2 V Gamma ....

T. Arbogast, C. N. Dawson, P. T. Keenan, M. F. Wheeler, and I. Yotov, Enhanced cell-centered finite differences for elliptic equations on general geometry, SIAM J. Sci. Comput., to appear.

....scheme. A variant of the mixed method, the expanded mixed method, has been developed for accurate and efficient treatment of irregular domains. The implementation and analysis of the method for single phase flow have been described in 4 MARY F. WHEELER AND IVAN YOTOV several previous works (see [6, 2, 3] for single block and [27, 5, 29] for multiblock domains) The original problem is transformed into a problem on a union of regular computational (reference) grids. The permeability after the mapping is usually a full tensor (except in some trivial cases) The mixed method could then be ....

....approximated by cell centered finite differences for the pressure, which is an efficient and highly accurate scheme [6] To simplify the presentation we will only describe here the rectangular reference case. For a definition of the spaces on logically rectangular and triangular grids, we refer to [2] (also see [24, 10] Let us denote the rectangular partition of Omega k by T hk , where h k is associated with the size of the elements. The lowest order Raviart Thomas spaces RT 0 [23] are defined on T hk by V hk = Phi v = v 1 ; v 2 ; v 3 ) vj E = ff 1 x 1 fi 1 ; ff 2 x 2 fi 2 ; ....

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T. Arbogast, C. N. Dawson, P. T. Keenan, M. F. Wheeler, and I. Yotov, Enhanced cellcentered finite differences for elliptic equations on general geometry, SIAM J. Sci. Comp. 19 (1998), no. 2, 404--425.

....for this application; however, other boundary conditions can be handled easily. A number of papers deal with the analysis and the implementation of mixed methods for our problem on conforming grids (see, e.g. 19] 21] 18] 4] 8] 6] 7] 11] 17] 22] 10] 12] 14] 3] [2] and the general references [9] 20] Mixed methods on nested locally refined grids are considered in [13] 15] but these techniques rely heavily on the fact that the grids are nested and cannot be extended directly to arbitrary non matching grids. Techniques have been developed to approximate ....

T. Arbogast, C. N. Dawson, P. T. Keenan, M. F. Wheeler, and I. Yotov, Enhanced cell-centered finite differences for elliptic equations on general geometry, SIAM J. Sci. Comput., 18 (1997). To appear. 1=h c kp \Gamma p h kM ku \Gamma u h kN kp \Gamma h kM

....rectangular meshes of quadrilateral type elements. Also, the computed velocities are not superconvergent on triangles, even for diagonal tensors. This motivates the choice of logically rectangular grids. Thomas [72] showed optimal convergence for both pressure and velocity on quadrilaterals. In [5], a modification of the expanded mixed method is considered for handling full tensors and general domains, leading to cellcentered finite difference schemes on logically rectangular and triangular grids. All computations are performed on a regular grid after mapping the problem to a reference ....

....macro hybrid form of the mixed method (see [11] has to be used for handling discontinuities. In this formulation, pressure Lagrange multipliers are added along the interfaces of discontinuities. This allows the pressure gradient to be approximated by a discontinuous function. Numerical results in [5] indicate that convergence is regained in this case. In Chapter 2 we present theoretical analysis for the macrohybrid form of the expanded mixed method for problems with piece wise smooth coefficients. The presence of faults imposes another very interesting problem for the numerical method. These ....

[Article contains additional citation context not shown here]

T. Arbogast, C. N. Dawson, P. T. Keenan, M. F. Wheeler, and I. Yotov, Enhanced cell-centered finite differences for elliptic equations on general geometry. To appear in SIAM J. Sci. Comp.

....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, C. N. Dawson, P. T. Keenan, M. F. Wheeler, and I. Yotov, Enhanced cell-centered finite differences for elliptic equations on general geometry. To appear in SIAM J. Sci. Comp.

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T. Arbogast, C. N. Dawson, P. T. Keenan, M. F. Wheeler and I. Yotov: ÒEnhanced cell-centered finite differences for elliptic equations on general geometry,Ó SIAM J. Sci. Comp., 19,

....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, C. N. Dawson, P. T. Keenan, M. F. Wheeler, and I. Yotov, Enhanced cellcentered finite differences for elliptic equations on general geometry, SIAM J. Sci. Comp., 19 (1998), pp. 404--425.

....considered for simplicity; however, other boundary conditions can be handled easily. A number of papers deal with the analysis and the implementation of mixed methods for our problem on conforming grids (see, e.g. 31] 34] 30] 5] 14] 12] 13] 20] 29] 35] 16] 21] 23] 4] [3] and the general references [15] 32] Mixed methods on nested locally refined grids are considered in [22] 24] but these techniques rely heavily on the fact that the grids are nested and cannot be extended directly to arbitrary non matching grids. In [26] Glowinski and Wheeler introduced two ....

....C A = 0 B B B B B x y 1 10 sin(6x) z 1 C C C C C A : 51) Fig. 2. Computed pressure on the second grid level. The pressure and the permeability are p(x; y; z) sin( x y z) K = 1 1 100 (x 2 y 2 z 2 ) We employ the expanded mixed method on curvilinear grids [3] and transform the problem to a computational problem with a modified full tensor coefficient on a union of rectangular grids. The multigrid algorithm is performed on the rectangular grids which allows for a trivial construction of coarse grids. The subdomain problems are solved efficiently by ....

T. Arbogast, C. N. Dawson, P. T. Keenan, M. F. Wheeler, and I. Yotov, Enhanced cell-centered finite differences for elliptic equations on general geometry, SIAM J. Sci. Comp., 19 (1998), pp. 404--425.

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Arbogast, T., Dawson, C.N., Keenan, P.T., Wheeler, M.F. and Yotov, I.: Enhanced cell-centered finite differences for elliptic equations on general geometry, to appear in SIAM J. Sci. Comp., 1997.

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