Convergence of the MPFA O-method on general grids

Abstract

The analysis of the Multi point flux approximation (MPFA) method has so far relied on the possibility of seeing it as a mixed finite element method for which the convergence is then established. This type of analysis has been successfully applied to triangles and quadrilaterals, lately also in the case of rough meshes.

Another well known conservative method, the mimetic finite difference method, has also traditionally relied on the analogy with a mixed finite element method to establish convergence. Recently however a new type of analysis proposed by Brezzi, Lipnikov and Shashkov (2005), permits to show convergence of the mimetic method on a general polyhedral mesh.

We propose to formulate the MPFA O-method in a mimetic finite difference framework, in order to extend the proof of convergence to polyhedral meshes. The formulation is useful to see the close relationship between the two different methods and to see how the differences lead to different strenghts. We pay special attention to the assumption needed for proving convergence by examining various cases in the section dedicated to numerical tests.