Stability of a combined finite element - finite volume discretization of convection-diffusion equations

We consider a time-dependent and a stationary convection-diffusion equation. These equations are approximated 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 nonstationary case, we use an implicit Euler approach for time discretization. This scheme is shown to be L2-stable uniformly with respect to the diffusion coefficient. In addition, it turns out that stability is unconditional in the time-dependent case. These results hold if the underlying grid satisfies a condition that is fulfilled, for example, by some structured meshes. (c) 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 402424, 2012