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

A class of a posteriori estimators is studied for the error in the maximum- norm of the gradient on single elements when the finite element method is u sed to approximate solutions of second order elliptic problems. The meshes are unstructured and, in particular, it is not assumed that there are any k nown superconvergent points. The estimators are based on averaging operator s which are approximate gradients, "recovered gradients", which are then co mpared to the actual gradient of the approximation on each element. Conditi ons are given under which they are asympotically exact or equivalent estima tors on each single element of the underlying meshes. Asymptotic exactness is accomplished by letting the approximate gradient operator average over d omains that are large, in a controlled fashion to be detailed below, compar ed to the size of the elements.