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A cell-centred finite volume approximation for second order partial derivative operators with full matrix on unstructured meshes in any space dimension

Robert Eymard·Thierry Gallouët·Raphaèle Herbin
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Abstract

Finite volume methods for problems involving second order operators with full diffusion matrix can be used thanks to the definition of a discrete gradient for piecewise constant functions on unstructured meshes satisfying an orthogonality condition. This discrete gradient is shown to satisfy a strong convergence property on the interpolation of regular functions, and a weak one on functions bounded for a discrete $H^1$ norm. To highlight the importance of both properties, the convergence of the finite volume scheme on a homogeneous Dirichlet problem with full diffusion matrix is proven, and an error estimate is provided. Numerical tests show the actual accuracy of the method.

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Robert Eymard, Thierry Gallouët, Raphaèle Herbin. 2005-05-06. A cell-centred finite volume approximation for second order partial derivative operators with full matrix on unstructured meshes in any space dimension. https://arxiv.org/abs/math/0505109

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