关键词：
boundary element method
adaptive algorithm
a posteriori error estimate
reliability
efficiency
摘要：
The article aims to provide a transparent introduction to and a state-of-the-art review on the mathematical theory of a posteriori error estimates for an operator equation Au =f on a one- (or two-) dimensional boundary surface (piece) Gamma. Symm's integral equation and a hypersingular equation serve as master examples for a boundary integral operator of the first kind. The non-local character of the involved pseudo-differential operator A and the non-local Sobolev spaces (of functions on Gamma) cause difficulties in the 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 localization arguments allow the derivation of reliable and/or efficient bounds eta = (Sigma (N)(j) = i eta (2)(j))(1/2). An error estimator eta is called efficient if C-1 eta less than or equal to E and reliable if E less than or equal to C-2 eta holds with multiplicative constants C-1 and C2, respectively, which are independent of underlying mesh-sizes, of data, or of the discrete and exact solution. The presented analysis of reliable and efficient 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 computable 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 multilevel method. Since the error estimators can be evaluated elementwise, they motivate error indicators eta (j), (better be named refinement- indicators) in adaptive mesh-refining algorithms. Although they perform very efficiently in practice, not much is rigorously known on the convergence of those schemes. ((C) 2001 Elsevier Science Ltd. All rights reser
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