Mathematical foundation of a posteriori error estimates and adaptive mesh-refining algorithms for boundary integral equations of the first kind

The article aims to provide a transparent introduction to and a state-of-th e-art review on the mathematical theory of a posteriori error estimates for an operator equation Au =f on a one- (or two-) dimensional boundary surfac e (piece) Gamma. Symm's integral equation and a hypersingular equation serv e as master examples for a boundary integral operator of the first kind. Th e non-local character of the involved pseudo-differential operator A and th e non-local Sobolev spaces (of functions on Gamma) cause difficulties in th e mathematical derivation of computable lower and upper error bounds for a discrete (known) approximation u(h) to the (unknown) exact solution u. If E denotes the norm of the error u - u(h) in a natural Sobolev norm, subtle l ocalization arguments allow the derivation of reliable and/or efficient bou nds eta = (Sigma (N)(j) = i eta (2)(j))(1/2). An error estimator eta is cal led efficient if C-1 eta less than or equal to E and reliable if E less tha n or equal to C-2 eta holds with multiplicative constants C-1 and C2, respe ctively, which are independent of underlying mesh-sizes, of data, or of the discrete and exact solution. The presented analysis of reliable and effici ent estimates is merely based on elementary calculus such as integration by parts or interchange of the order of integration along the curve Gamma. Four examples of residual-based partly reliable and partly efficient comput able error estimators eta (j) are discussed such as the weighted residuals on an element Gamma (j), the localized residual norm on Gamma (j), the norm of a solution of a certain local problem, or the correction in a multileve l method. Since the error estimators can be evaluated elementwise, they motivate erro r indicators eta (j), (better be named refinement- indicators) in adaptive mesh-refining algorithms. Although they perform very efficiently in practic e, not much is rigorously known on the convergence of those schemes. ((C) 2 001 Elsevier Science Ltd. All rights reserved.