KOEBBE, J., A Computationally Efficient Modification of Mixed Finite-Element Methods for Flow Problems with Full Transmissiity Tensors, Numerical Methods for Partial Differential Equations, Vol. 9, pp. 339--355, 1993.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....use of the expanded mixed formulation briefly in a practical setting in [33] A similar formulation was considered by Chen [11] to approximate a nonlinear problem, using only the BDM spaces [9] He also presented a convergence analysis (see also [12] but did not discuss implementation. Koebbe [22] used the expanded mixed formulation to solve problems with a tensor coefficient. He was concerned with implementation, but he did not attempt to obtain a finite difference stencil. Rather, he solved a saddle point problem. We consider a more general set of test and trial functions than either ....

....the expanded mixed formulation to solve problems with a tensor coefficient. He was concerned with implementation, but he did not attempt to obtain a finite difference stencil. Rather, he solved a saddle point problem. We consider a more general set of test and trial functions than either [11] or [22], since they both took V = V . The rest of the paper is organized as follows. In Section 2 we formulate the discrete approximation of (1.3) Stability and solvability are shown, and a convergence theorem is given in Section 3. The cell centered finite difference stencil for the pressure on ....

J. Koebbe, A computationally efficient modification of mixed finite element methods for flow problems with full transmissivity tensors, Numer. Meth. for PDE's, 9 (1993), pp. 339--355.

....[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 be eliminated and the method reduces to a positive definite, cell centered ....

....v 2 H(div; Omega Gamma ; ffp; w) r Delta u; w) f; w) w 2 L 2( Omega Gamma ; 3.3.3c) hu Delta ; i Gamma N = hg N ; i Gamma N ; 2 H 1=2 ( Gamma N ) 3.3.3d) In the usual expanded formulation, G is taken to be the identity. In the recently introduced expanded mixed method [37, 25, 10, 2], we need an additional finite element space V h such that V h V h (L 2( Omega Gamma9 d . Let V h = spanf v m : m = 1; N V g: In our modification to the expanded mixed method, we seek U 2 V h , U 2 V h , mixed finite element methods on general geometry 9 P 2 W ....

J. Koebbe, A computationally efficient modification of mixed finite element methods for flow problems with full transmissivity tensors, Numer. Meth. for PDE's, 9 (1993), pp. 339-- 355.

....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, ....

J. Koebbe, A computationally efficient modification of mixed finite element methods for flow problems with full transmissivity tensors, Numer. Meth. for PDE's, 9 (1993), pp. 339--355.

....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 out for the first time for second order differential problems in [8] and [9] and for fourth order differential problems in this paper. II. NOTATION AND PRELIMINARIES In this section we review some known results about ....

J. Koebbe, A computationally efficient modification of mixed finite element methods for flow problems with full transmissivity tensors, Numer. Meth. for PDE's 9 (1993), 339--355.

....while the convergence results and the pressure superconvergence results can be easily generalized for a full tensor, this is not the case for the superconvergence in the velocity. Recently, a variant of the mixed finite element method, which we call the expanded mixed method , has been introduced [75, 52, 25, 9]. In [9] full tensor coefficients are handled efficiently on rectangles by deriving an equivalent cell centered finite difference scheme for the pressure, which generalizes the result of Russell and Wheeler [70] for diagonal tensors. Moreover, superconvergence for the velocity is obtained at ....

J. Koebbe, A computationally efficient modification of mixed finite element methods for flow problems with full transmissivity tensors, Numer. Meth. for PDE's, 9 (1993), pp. 339--355.

....fine to coarse data [26] or when mapping a rectangular grid into a logically rectangular grid [7] When the tensor is full it is not possible to derive a finite difference scheme equivalent to the mixed method. Recently, methods have been developed to handle a full, possibly noninvertible tensor [8, 19, 44]. In particular, Arbogast, Wheeler and Yotov have analyzed the expanded mixed finite element method [8] This method simultaneously approximates the pressure, its gradient and the flux. Arbogast, Wheeler and Yotov showed that for the lowest order Raviart Thomas Nedelec space on parallelepipeds, a ....

J. Koebbe, A computationally efficient modification of mixed finite element methods for flow problems with full transmissivity tensors, Numer. Meth. for PDE's, 9 (1993), pp. 339--355.

....use of the expanded mixed formulation briefly in a practical setting in [32] A similar formulation was considered by Chen [11] to approximate a nonlinear problem, using only the BDM spaces [9] He also presented a convergence analysis (see also [12] but did not discuss implementation. Koebbe [21] used the expanded mixed formulation to solve problems with a tensor coefficient. He was concerned with implementation, but he did not attempt to obtain a finite difference stencil. Rather, he solved a saddle point problem. We consider a more general set of test and trial functions than either ....

....the expanded mixed formulation to solve problems with a tensor coefficient. He was concerned with implementation, but he did not attempt to obtain a finite difference stencil. Rather, he solved a saddle point problem. We consider a more general set of test and trial functions than either [11] or [21], since they both took V = V . The rest of the paper is organized as follows. In Section 2 we formulate the discrete approximation of (1.3) Stability and solvability are shown, and a convergence theorem is given in Section 3. The cell centered finite difference stencil for the pressure on ....

J. Koebbe, A computationally efficient modification of mixed finite element methods for flow problems with full transmissivity tensors, Numer. Meth. for PDE's, 9 (1993), pp. 339-- 355.

....29] The relationship between the mixed method on rectangular meshes and cell centered finite differences was first established in [29] provided that K in (1. 1) is a scalar or a diagonal matrix, and later for general tensor K in [1] for a variant of the mixed method, the expanded mixed method [34, 22, 7, 1], again provided that the mesh is rectangular. If one uses the RT 0 space and applies appropriate quadrature rules, the velocity unknowns can be eliminated and the method reduces to a positive definite, cell centered finite difference method for the pressure p with a stencil of 9 points if d = 2 ....

....I ) 2.1d) This is expanded from the standard mixed variational form in that we have introduced a symmetric positive definite tensor field G and an additional unknown (2.2) u = GammaG Gamma1 rp; that represents an adjusted gradient . In the recently introduced expanded formulation [34, 22, 7, 1], G is the identity. If instead G = K Gamma1 , u = u and one recovers the standard mixed method formulation. Later we define G based on the local geometry. 3. An expanded mixed method. Let fE h gh 0 be a regular family of finite element partitions of Omega [9] where h is the maximal element ....

J. Koebbe, A computationally efficient modification of mixed finite element methods for flow problems with full transmissivity tensors, Numer. Meth. for PDE's, 9 (1993), pp. 339--355.

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KOEBBE, J., A Computationally Efficient Modification of Mixed Finite-Element Methods for Flow Problems with Full Transmissiity Tensors, Numerical Methods for Partial Differential Equations, Vol. 9, pp. 339--355, 1993.

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