Schatz  has recently proven sharply localized, or weighted, maximum norm estimates for Galerkin methods on unstructured meshes. These estimates generalize previous maximum norm stability results by showing that the higher the order of polynomial used in a Galerkin method, the more local the resulting approximation. We present analogous results for a mixed method for linear elliptic problems. Our estimates are valid for the vector variable, scalar variable, and a superconvergent postprocessed approximation to the scalar variable, and hold for all of the typical element spaces used in this context. We also comment on the best choice of element space. In particular, our estimates indicate that the lowest order Raviart-Thomas elements give a localized approximation to the vector variable. In contrast, the lowest order Brezzi-Douglas-Marini (BDM) elements approximate the vector variable to one higher order than the Raviart-Thomas elements but do not appear to yield a localized approximation.
|||Pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids. I. Global estimates. Math. Comp. 67 (1998), no. 223, 877-899.|