Efficiency and Optimality of Some Weighted-Residual Error Estimator for Adaptive 2D Boundary Element Methods

Markus Aurada 1 , Michael Feischl 2 , Thomas Führer 3 , Michael Karkulik 4 ,  and Dirk Praetorius 5
  • 1 Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstr. 8–10, 1040 Wien, Austria
  • 2 Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstr. 8–10, 1040 Wien, Austria
  • 3 Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstr. 8–10, 1040 Wien, Austria
  • 4 Departamento de Matemáticas, Avenida Vicuña Mackenna 4860, Santiago, Chile
  • 5 Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstr. 8–10, 1040 Wien, Austria

Abstract.

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.

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CMAM considers original mathematical contributions to computational methods and numerical analysis with applications mainly related to PDEs. The journal is interdisciplinary while retaining the common thread of numerical analysis, readily readable and meant for a wide circle of researchers in applied mathematics.

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