Carsten Carstensen, Birgit Faermann:

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-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 nonlocal character of the involved pseudodifferential operator A and the nonlocal 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 = (SUM[j=1,...,N] eta_j²)^1/2. An error estimator eta is called efficient if C₁ eta <= E resp. reliable if E <= C₂ eta holds with multiplicative constants C₁ resp. C₂ 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 efficient in practice, not much is rigorously known on the convergence of those schemes.

Mathematics Subject Classification (1991): 65N38, 65N15, 65R20, 45L10

Keywords: boundary element method, adaptive algorithm, a posteriori error estimate, reliability, efficiency

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[Thu Feb 19 18:56:35 2009]

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