T. Gallouet et al., Error estimates on the approximate finite volume solution of convection diffusion equations with general boundary conditions, SIAM J NUM, 37(6), 2000, pp. 1935-1972
We study here the convergence of a finite volume scheme for a diffusion con
vection equation on an open bounded set of R-d (d = 2 or 3) for which we co
nsider Dirichlet, Neumann, or Robin boundary conditions. We consider unstru
ctured meshes which include Voronoi or triangular meshes; we use for the di
ffusion term an "s points" (where s is the number of sides of each cell) fi
nite volume scheme and for the convection term an upstream finite volume sc
heme. Assuming the exact solution at least in H-2 we prove error estimates
ina discrete H-0(1) norm of order the size of the mesh. Discrete Poincare i
nequalities then allow one to prove error estimates in the L-2 norm.