THE INFLUENCE AND SELECTION OF SUBSPACES FOR A-POSTERIORI ERROR ESTIMATORS

The element residual method for a posteriori error estimation is analy zed for degree p finite element approximation on quadrilateral element s. The influence of the choice of subspace used to solve the element r esidual problem is studied. It is shown that the resulting estimators will be consistent (or asymptotically exact) for all p > 1 if and only if the mesh is parallel. Moreover, even if the mesh consists of recta ngles, then the estimators can be inconsistent when p = 1. The results provide concrete guidelines for the selection of a posteriori error e stimators and establish the limits of their performance. In particular , the use of the element residual method for high orders of approximat ion (such as those arising in the h-p version finite element method) i s vindicated. The mechanism behind the rather poor performance of the estimators is traced back to the basic formulation of the residual pro blem. The investigations reveal a deficiency in the formulation, leadi ng, as it does, to spurious modes in the true solution of the residual problem, The recommended choice of subspaces may be viewed as being s ufficient to guarantee that the spurious modes are filtered out from t he approximate solution while at the same time retaining a sufficient degree of approximation to represent the true modes.