The influence and selection of subspaces for a posteriori error estimators

Summary.

The element residual method for a posteriori error estimation is analyzed for degree $p$ finite element approximation on quadrilateral elements. The influence of the choice of subspace used to solve the element residual 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 rectangles, then the estimators can be inconsistent when $p=1$. The results provide concrete guidelines for the selection of a posteriori error estimators and establish the limits of their performance. In particular, the use of the element residual method for high orders of approximation (such as those arising in the $h$-$p$ version finite element method) is vindicated. The mechanism behind the rather poor performance of the estimators is traced back to the basic formulation of the residual problem. The investigations reveal a deficiency in the formulation, leading, as it does, to spurious modes in the true solution of the residual problem. The recommended choice of subspaces may be viewed as being sufficient to guarantee that the spurious modes are filtered out from the approximate solution while at the same time retaining a sufficient degree of approximation to represent the true modes.