Asymptotically exact a posteriori estimators for the pointwise gradient error on each element in irregular meshes. Part 1: A smooth problem and globally quasi-uniform meshes

Abstract

A class of a posteriori estimators is studied for the error in the maximum-norm of thegradient on single elements when the finite element method is used toapproximate solutions of second order elliptic problems. The meshes areunstructured and, in particular, it is not assumed that there are any knownsuperconvergent points. The estimators are based on averaging operatorswhich are approximate gradients, recovered gradients, which arethen compared to the actual gradient of the approximation on eachelement. Conditions are given under which they are asympotically exactor equivalent estimators on each single element of the underlyingmeshes. Asymptotic exactness is accomplished by letting the approximategradient operator average over domains that are large, in a controlledfashion to be detailed below, compared to the size of the elements.