About Mixed Finite Elements and their application to a priori and a posteriori error estimation for a class of 2-d and n-d Finite Volume Schemes

A class of cell-centered Finite Volume (F.V.) Schemes can be efficiently analyzed in the framework of Mixed Finite Element (M.F.E.) Theory. For the model problem, the F.V. equations in the scalar field can be obtained from the dual mixed system using Raviart-Thomas element and a well-chosen numerical integration formula ("mass lumping").

This techniques is a good way for exhibiting a priori estimates in natural norms for the F.V. Schemes, the proof of which using only classical results of M.F.E. theory. It has been possible for 2-d triangular cells (but still not for n-d simplicial ones!), and for n-d rectangular cells.

This techniques allows also to construct a posteriori estimators for F.V. Schemes, which is important for F.V. codes. The method is based on a generalization to the case with numerical integration of Arnold-Brezzi results which associate to the mixed system a resolving nonconforming F.E. problem. We emphasize that there are significant differences between triangles and rectangles in proving the results.

Most of the present results have been obtained in collaboration with A.Agouzal, J.Baranger, J.Olaiz and F.Oudin.