This paper studies the combination of the mixed covolume method with the expanded mixed element method for elliptic problems defined on a rectangle

${\Omega}$. The lowest order Raviart-Thomas element on a rectangular grid is used. The analytical results include the uniqueness of the discrete solution if the mesh is sufficiently fine, the first order convergence in

${L}^{2}\left({\Omega}\right)$ and superconvergence results for discrete norms. Second order superconvergence can be achieved if the grid becomes sufficiently fine near the boundary. The method can be transformed into a scalar finite difference scheme for one of the variables if an appropriate quadrature rule is used. Simple numerical examples support the analytical results.