L²-stability of a finite element – finite volume discretization of convection-diffusion-reaction equations with nonhomogeneous mixed boundary conditions

¹ Universitédu Littoral Côte d’Opale, Laboratoire de mathématiques pures et appliquées Joseph Liouville, 62228 Calais, France.
Paul.Deuring@lmpa.univ-littoral.fr
² UniversitéParis-Est Marne-la-Vallée, 5 boulevard Descartes, 77454 Marne-la-Vallée, France.

Received: 9 September 2015
Revised: 18 March 2016
Accepted: 30 May 2016

Abstract

We consider a time-dependent and a steady linear convection-diffusion-reaction equation whose coefficients are nonconstant. Boundary conditions are mixed (Dirichlet and Robin−Neumann) and nonhomogeneous. Both the unsteady and the steady problem are approximately solved by a combined finite element – finite volume method: the diffusion term is discretized by Crouzeix−Raviart piecewise linear finite elements on a triangular grid, and the convection term by upwind barycentric finite volumes. In the unsteady case, the implicit Euler method is used as time discretization. This scheme is shown to be unconditionally L²-stable, uniformly with respect to diffusion, except if the Robin−Neumann boundary condition is inhomogeneous and the convective velocity is tangential at some points of the Robin−Neumann boundary. In that case, a negative power of the diffusion coefficient arises. As is shown by a counterexample, this exception cannot be avoided.