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

¹ School of Mathematical Sciences, Monash University, 3800 Victoria, Australia
jerome.droniou@monash.edu
² Laboratoire d’Analyse et de Mathématiques Appliquées, Université Paris-Est, UMR 8050, 5 boulevard Descartes, Champs-sur-Marne, 77454 Marne-la-Vallée cedex 2, France
Robert.Eymard@univ-mlv.fr
³ Laboratoire d’Analyse Topologie et Probabilités, UMR 6632, Université d’Aix-Marseille, 39 rue Joliot Curie, 13453 Marseille, France Raphaele
Herbin@latp.univ-mrs.fr

Received: 15 April 2015
Revised: 5 October 2015

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.