A cell-centred finite-volume approximation for anisotropic diffusion operators on unstructured meshes in any space dimension. (English) Zbl 1093.65110

Eymard, R.; Gallouët, T.; Herbin, R.

IMA J. Numer. Anal. 26, No. 2, 326-353 (2006).

A discrete gradient for piecewise constant functions is constructed. This discrete gradient exhibits several advantages: it is easy and cheap to compute, and it provides a simple scheme for the approximation of anisotropic convection-diffusion problems. Main result: The authors show a weak convergence property of the discrete gradient to the limit of the sequence of functions, together with a consistency property, both leading to the strong convergence of the discrete solution and of its discrete gradient in the case of a Dirichlet problem with full matrix diffusion. The precise proof that the discrete gradient to satisfy a strong convergence property for the interpolation of regular functions, and a weak one for functions bounded in an \(H^1\) norm is proposed. Numerical tests show the actual accuracy of authors’ method.

Reviewer: Jan Lovíšek (Bratislava)

MSC:

65N30	Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N12	Stability and convergence of numerical methods for boundary value problems involving PDEs
35J05	Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation

Keywords:

anisotropic diffusion; finite-volume method; discrete gradient; weak and strong convergence; homogeneous Dirichlet problem; connection-diffusion problem