Chapter

# Benchmark 3D: the VAG scheme

12/2010; DOI:10.1007/978-3-642-20671-9_99

ABSTRACT Let Ωbe a bounded open domain of

\mathbbR3\mathbb{R}^3
let
feL2(W)f\epsilon\rm L^2(\Omega)
and let
L\Lambda
be a measurable function from Ω to the set

m3(\mathbbR) of 3 3m^3(\mathbb{R)} of 3 \times 3
matrices, such that for a.e.

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##### Article: Benchmark on discretization schemes for anisotropic diffusion problems on general grids
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ABSTRACT: We present here a number of test cases and meshes which were designed to form a benchmark for finite volume schemes and give a summary of some of the results which were presented by the participants to this benchmark. We address a two-dimensional anisotropic diffusion problem, which is discretized on general, possibly non-conforming meshes. In most cases, the diffusion tensor is taken to be anisotropic, and at times heterogeneous and/or dis-continuous. The meshes are either triangular or quadrangular, and sometimes quite distorted. Several methods were tested, among which finite element, discontinous Galerkin, cell centred and vertex centred finite volume methods, discrete duality finite volume methods, mimetic meth-ods. The results given by the participants to the benchmark range from the number of unknowns, the errors on the fluxes or the minimum and maximum values and energy, to the order of con-vergence (when available).
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##### Chapter: Multiphase Flow in Porous Media Using the VAG Scheme
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ABSTRACT: We present the use of the Vertex Approximate Gradient scheme for the simulation of multiphase flow in porous media. The porous volume is distributed to the natural grid blocks and to the vertices, hence leading to a new finite volume mesh. Then the unknowns in the control volumes may be eliminated, and a 27-point scheme results on the vertices unknowns for a hexahedral structured mesh. Numerical results show the efficiency of the scheme in various situations, including miscible gas injection. Keywordstwo-phase flow in porous media-vertex approximate gradient scheme-reservoir simulation
12/2010: pages 409-417;

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