T. Arbogast, C. N. Dawson, P. T. Keenan, M. F. Wheeler, and I. Yotov, Implementation of mixed finite element methods for elliptic equations on general geometry, (in preparation).  Home/Search   Document Details and Download   Summary   Related Articles   Check

....in the aerospace industry [144] and has recently been extended to problems in petroleum reservoir engineering [1] This method was used in the GCT transport module. The original physical problem is mapped to the computational domain, giving a similar problem with a modified tensor coefficient [2, 6, 9, 14]. The results in [13] were used to transform the mixed method into a simple cell centered finite difference method. The resulting scheme is an approximation to the physical problem, and it is locally mass conservative. The mixed finite elements in the physical domain are not quadrilaterals or ....

T. Arbogast, C. N. Dawson, P. T. Keenan, M. F. Wheeler, and I. Yotov, Implementation of mixed finite element methods for elliptic equations on general geometry, (in preparation).

....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) ....

....u h ; w) f; w) w 2 W h ; 2.13) hu h Delta ; i Gamma N = hg N ; i Gamma N ; 2 N h : 2. 14) Optimal convergence for kp Gamma p h k 0 , k Gamma h k Gamma1=2 , ku Gamma u h k 0 , k u Gamma u h k 0 , and kr Delta (u Gamma u h )k 0 in all known mixed spaces has been shown in [4, 1] under the assumption of smoothness of K and M . Here we present the results for the lowest order Raviart Thomas spaces [44, 42] Theorem 2.1. For the expanded mixed method (2.11) 2.14) on curved elements, there exists a constant C, independent of h and dependent on Omega ; kpk 2 ; kuk 1 ; ....

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T. Arbogast, C. N. Dawson, P. T. Keenan, M. F. Wheeler, and I. Yotov, Implementation of mixed finite element methods for elliptic equations on general geometry. Submitted.

....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 ....

....for creating F and its Jacobian matrix. Our work here 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 ....

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Arbogast, T., Dawson, C., Keenan, P., Wheeler, M.F., and Yotov, I.: "Implementation of mixed finite element methods for elliptic equations on general geometry" (in preparation).

....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) ....

....h on u h . c) Stencil for the pressure p h . 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, Implementation of mixed finite element methods for elliptic equations on general geometry, to appear.

....: 30 Chapter 1 Introduction This manual describes the user interface to RUF, the Rice Unstructured Flow program. RUF solves scalar linear second order elliptic equations on general unstructured meshes in two or three space dimensions using mixed finite element methods [1, 2, 3]. It is applicable to steady state flow calculations in porous media, such as arise in petroleum reservoir simulation and groundwater contaminant modeling. Extensions to nonlinear time dependent systems such as arise in multi phase flow and transport are under development. Version 2.3 uses the ....

....method when that is viewed as a mixed method with quadrature. The programs are otherwise identical and build on a substantial C library of tools for partial differential equations, general geometry, linear algebra and user interfaces. The various numerical methods are defined in detail in [1, 2, 3], which also presents numerical examples and explains which methods are preferred. RUF 1.0 [7] included versions for the saddle point formulation of the mixed finite element method, which has been dropped in subsequent versions because it is too inefficient. The stencil version in RUF 1.0 is now ....

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Arbogast, T., Dawson, C., Keenan, P. T., Wheeler, M. F., and Yotov, I., Implementation of Mixed Finite Element Methods for Elliptic Equations on General Geometry, Dept. of Computational and Applied Mathematics Tech. Report #95--??, Rice University, 1995, and To Appear.

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