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 for example the modeling of contaminant transport in porous media. The diffusion term, which can be anisotropic and heterogeneous, is discretized using a recently developed hybrid mimetic mixed framework. We construct a family of discretizations for the convection term, which uses the hybrid interface unknowns. We consider a wide range of unstructured possibly nonmatching polyhedral meshes in arbitrary space dimension. The scheme is fully implicit in time, it is locally conservative and robust with respect to the Peclet number. We obtain a convergence result based upon a priori estimates and the Frechet---Kolmogorov compactness theorem. We implement the scheme both in two and three space dimensions and compare the numerical results obtained with the upwind and the centered discretizations of the convection term numerically.