Finite volume schemes for diffusion equations: introduction to and review of modern methods

TL;DR

This paper reviews modern finite volume schemes for diffusion equations on generic meshes, emphasizing their stability, convergence, and physical relevance, especially under anisotropic conditions, based on recent literature developments.

Contribution

It provides an introduction to and a comprehensive review of finite volume methods, highlighting key properties like coercivity and minimum-maximum principles for diffusion equations.

Findings

Finite volume schemes ensure stability and convergence.

Methods maintain physical bounds under anisotropy.

Review covers recent advances in the field.

Abstract

We present Finite Volume methods for diffusion equations on generic meshes, that received important coverage in the last decade or so. After introducing the main ideas and construction principles of the methods, we review some literature results, focusing on two important properties of schemes (discrete versions of well-known properties of the continuous equation): coercivity and minimum-maximum principles. Coercivity ensures the stability of the method as well as its convergence under assumptions compatible with real-world applications, whereas minimum-maximum principles are crucial in case of strong anisotropy to obtain physically meaningful approximate solutions.