Some remarks about the hierarchical a posteriori error estimate

The saturation assumption is widely used in a posteriori error analysis of finite element methods. It asserts, in its simplest form, that the solution can be approximated asymptotically better with quadratic than with linear finite elements. In this article, we show that a simple modification of this "hypothesis" is valid, and the proof of many authors can be made rigorous with this simple modification. We prove also the robustness of the estimator for a singularly perturbed problem. (C) 2004 Wiley Periodicals, Inc.