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

TL;DR

This paper introduces a robust finite volume method on general, possibly non-matching meshes for degenerate parabolic convection-reaction-diffusion equations, ensuring convergence and applicability to complex geometries.

Contribution

It develops a novel hybrid finite volume scheme with a partially upwind convection discretization for complex, unstructured meshes, and proves its convergence.

Findings

The scheme is fully implicit, locally conservative, and robust across Péclet numbers.

Convergence is established using a priori estimates and the Fréchet–Kolmogorov theorem.

Applicable to anisotropic, heterogeneous diffusion in arbitrary dimensions.

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.