Finite volume box schemes and mixed methods

We present the numerical analysis on the Poisson problem of two mixed Petrov-Galerkin finite volume schemes for equations in divergence form div phi>(*) over bar * (u, delu) = f. The first scheme, which has been introduced in [22], is a generalization in two dimensions of Keller's box-scheme. The second scheme is the dual of the first one, and is a cell-centered scheme for u, and the flux phi. For the first scheme, the two trial finite element spaces are the nonconforming space of Crouzeix-Raviart for the primal unknown u, and the div-conforming space of Raviart-Thomas for the flux phi. The two test spaces are the functions constant per cell both for the conservative and for the flux equations. We prove an optimal second order error estimate for the box scheme and we emphasize the link between this scheme and the post-processing of Arnold and Brezzi of the classical mixed method.