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M. Aurada, M. Feischl, T. Führer, M. Karkulik, D. Praetorius:
"Efficiency and optimality of some weighted-residual error estimator for adaptive 2D boundary element methods";
Computational Methods in Applied Mathematics, 13 (2013), 03; S. 305 - 332.

We prove convergence and quasi-optimality of a lowest-order adaptive boundary element method for a weakly-singular integral equation in 2D. The adaptive mesh refinement is driven by the weighted-residual error estimator. By proving that this estimator is not only reliable, but under some regularity assumptions on the given data also efficient on locally refined meshes, we characterize the approximation class in terms of the Galerkin error only. In particular, this yields that no adaptive strategy can do better, and the weighted-residual error estimator is thus an optimal choice to steer the adaptive mesh-refinement. As a side result, we prove a weak form of the saturation assumption.

boundary element method, weakly-singular integral equation, a posteriori error estimate, adaptive algorithm, convergence, optimality

http://dx.doi.org/10.1515/cmam-2013-0010