Interior superconvergence in mortar and non-mortar mixed finite element methods on non-matching grids

We establish interior velocity superconvergence estimates for mixed finite element approximations of second order elliptic problems on non-matching rectangular and quadrilateral grids. Both mortar and non-mortar methods for imposing the interface conditions are considered. In both cases it is shown that a discrete L2-error in the velocity in a compactly contained subdomain away from the interfaces converges of order O(h1/2) higher than the error in the whole domain. For the non-mortar method we also establish pressure superconvergence, which is needed in the velocity analysis. Numerical results are presented in confirmation of the theory.