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 strong convergence property for the interpolation of regular functions, and a weak one for functions bounded in a discrete H
¹
-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.