Analysis tools for finite volume schemes

Titre de la revue : Acta Mathematica Universitatis Comenianae
Volume : 76
N° : 1
Pagination : 111-136
Date de publication : 01/02/2007

This paper is devoted to a review of the analysis tools which have been developed for the the mathematical study of cell centred finite volume schemes in the past years. We first recall the general principle of the method and give some simple examples. We then explain how the analysis is performed for elliptic equations and relate it to the analysis of the continuous problem; the lack of regularity of the approximate solutions is overcome by an estimate on the translates, which allows the use of the Kolmogorov theorem in order to get compactness. The parabolic case is treated with the same technique. Next we introduce a co-located scheme for the incompressible Navier-Stokes equations, which requires the definition of some discrete derivatives. Here again, we explain how the continuous estimates can guide us for the discrete estimates. We then give the basic ideas of the convergence analysis for non linear hyperbolic conservation laws, and conclude with an overview of the recent domains of application.