Superconvergence and the use of the residual as an error estimator in the BEM. II: Collocation, numerical integration and error indicators

We show for a variety of integral equations that the residual can be used as an error estimator provided the Sloan iterate of the approximation superconverges. This generalizes a result given by Geng el al. [J. Acoust. Soc. Am. 100 (1996) 355]. When the solution technique is Galerkin's method we show that the superconvergence of the Sloan iterate can be established under quite general conditions. For collocation this is more difficult and we discuss a generalization of a result of Brunner [J. Comp. Appl. Math. 67 (1996) 185] for doing this. Using ideas of Schulz [Uher lokale and globale Fehlergsehatzungen fur adaptive randelment Methoden, PhD thesis, Mathematisches Institute A, University of Stuttgart, 1997] it is shown how to localize these results to provide asymptotically exact local error indicators. It is also shown that it is important to consider the effect of numerical integration errors. as such errors can destroy superconvergence. (C) 2001 Elsevier Science Ltd. All rights reserved.