A cell-centred finite-volume approximation for anisotropic diffusion operators on unstructured meshes in any space dimension

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 stron...

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Bibliographic Details
Published in:IMA journal of numerical analysis 2006-04, Vol.26 (2), p.326-353
Main Authors: Eymard, R., Gallouèˆt, T., Herbin, R.
Format: Article
Language:English
Subjects:
anisotropic diffusion
convergence analysis
discrete gradient
Exact sciences and technology
finite-volume methods
Mathematics
Numerical analysis
Numerical analysis. Scientific computation
Numerical approximation
Partial differential equations, boundary value problems
Sciences and techniques of general use
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Summary: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 for the interpolation of regular functions, and a weak one for functions bounded in a discrete H1-norm. To highlight the importance of both properties, the convergence of the finite-volume scheme for 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.
ISSN:0272-4979
1464-3642
DOI:10.1093/imanum/dri036