Pointwise Error Estimates for First-Order div Least-Squares Finite Element Methods and Applications to Superconvergence and A Posteriori Error Estimators

Abstract

First-order div least-squares finite element methods for second-order elliptic partial differential equations are considered. While there has been significant progress in error estimates for the methods, to the best of our knowledge these estimates are based on the global norm of the error. In this paper, we provide highly localized pointwise error estimates for the primary variable u by establishing that the least-squares solutions are locally higher-order perturbations of the standard Galerkin solutions. As elementary consequences, we identify the superconvergent points. The set of superconvergent points for the primary function of the least-squares solution is the same as that of the standard Galerkin solution. Also, we present a class of average-type a posteriori error estimators for the method, and conditions are given under which they are asymptotically exact or equivalent estimators in the maximum-norm on each single element of the underlying mesh.