This talk will discuss superconvergence and its application to posteriori error estimation for finite element solutions of partial differential equations. In particular, we are interested in projection methods defined locally on subdomains. The projection method is a post- processing procedure that constructs new approximations by using the method of least-squares. This procedure has been proved to produce new approximations with superconvergence on quasi-uniform meshes. The existing results rely on a global a prior regularity of the adjoint problem. The goal of this talk is to establish some interior superconvergence estimates in the L2 and L∞ norms for local projections of the Galerkin finite element solutions. The results have two prominent features. First, they are established for any quasi-uniform meshes

which are of practical importance in scientific computing. Second, they are derived on the basis of local properties of the domain and the solution for the second order elliptic problem. As a result, the global a priori regularity on the adjoint problem is no longer required in the superconvergence. Therefore, the result of this paper can be employed to provide local and useful a posteriori error estimators in practical scientific computing. This is a joint work with Professor Hongsen Chen at the University of Wyoming.