Some New Local Error Estimates in Negative Norms with an Application to Local a Posteriori Error Estimation

Abstract

Here we survey some previously published results and announce some that have been newly obtained. We first review some of the results in [3] on estimates for the finite element error at a point. These estimates and analogous ones in [4] and [7] have been applied to problems in a posteriori estimates [2], [8], superconvergence [5] and others [9], [10]. We then discuss the extension of these estimates to local estimates in $L_∞$ based negative norms. These estimates have been newly obtained and are applied to the problem of obtaining an asymptotically exact a posteriori estimator for the maximum norm of the solution error on each element.