Error Estimates on the Approximate Finite Volume Solution of Convection Diffusion Equations with General Boundary Conditions

Abstract

We study here the convergence of a finite volume scheme for a diffusion convection equation on an open bounded set of $\R^\dim$ ($\dim=2$ or 3) for which we consider Dirichlet, Neumann, or Robin boundary conditions. We consider unstructured meshes which include Voronoi or triangular meshes; we use for the diffusion term an "s points" (where s is the number of sides of each cell) finite volume scheme and for the convection term an upstream finite volume scheme. Assuming the exact solution at least in H2 we prove error estimates in a discrete $H^1_0$ norm of order the size of the mesh. Discrete Poincaré inequalities then allow one to prove error estimates in the L2 norm.