A New Post-Processing Technique for Finite Element Methods with $L^2$-Superconvergence

Abstract

A simple post-processing technique for finite element methods with $L$²-superconvergence is proposed. It provides more accurate approximations for solutions of two- and three-dimensional systems of partial differential equations. Approximate solutions can be constructed locally by using finite element approximations $u$_$h$ provided that $u$_$h$ is superconvergent for a locally defined projection $\widetilde{P}$_$h$$u$. The construction is based on the least-squares fitting algorithm and local $L$²-projections. Error estimates are derived and numerical examples illustrate the effectiveness of this approach for finite element methods.