Gradient schemes: generic tools for the numerical analysis of diffusion equations

TL;DR

This paper develops a unified framework called gradient schemes for analyzing diffusion equations, providing generic tools to verify properties of classical and new numerical methods on polytopal meshes.

Contribution

It introduces new generic tools within the gradient scheme framework, enabling systematic analysis and validation of diffusion schemes, including mass lumping and barycentric condensation techniques.

Findings

Classical schemes are confirmed as gradient schemes.

New tools facilitate analysis of mass lumping processes.

A discrete functional analysis toolbox for polytopal meshes is developed.

Abstract

The gradient scheme framework is based on a small number of properties and encompasses a large number of numerical methods for diffusion models. We recall these properties and develop some new generic tools associated with the gradient scheme framework. These tools enable us to prove that classical schemes are indeed gradient schemes, and allow us to perform a complete and generic study of the well-known (but rarely well-studied) mass lumping process. They also allow an easy check of the mathematical properties of new schemes, by developing a generic process for eliminating unknowns via barycentric condensation, and by designing a concept of discrete functional analysis toolbox for schemes based on polytopal meshes.