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. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 402–424, 2012