A finite volume method on general meshes for a degenerate parabolic convection-reaction-diffusion equation

Abstract: We propose a finite volume method on general meshes for the discretization of a degenerate parabolic convection-reaction-diffusion equation. Equations of this type arise in many contexts, such as the modeling of contaminant transport in porous media. We discretize the diffusion term, which can be anisotropic and heterogeneous, via a hybrid finite volume scheme. We construct a partially upwind scheme for the convection term. We consider a wide range of unstructured possibly non-matching polygonal meshes in arbitrary space dimension. The only assumption on the mesh is that the volume elements must be star-shaped. The scheme is fully implicit in time, it is locally conservative and robust with respect to the P\'eclet number. We obtain a convergence result based upon a priori estimates and the Fr\'echet--Kolmogorov compactness theorem.