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Adaptive boundary element methods for optimal convergence of point errors

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Abstract

One particular strength of the boundary element method is that it allows for a high-order pointwise approximation of the solution of the related partial differential equation via the representation formula. However, the high-order convergence and hence accuracy usually suffers from singularities of the Cauchy data. We propose two adaptive mesh-refining algorithms and prove their quasi-optimal convergence behavior with respect to an a posteriori computable bound for the point error in the representation formula. Numerical examples for the weakly-singular integral equations for the 2D and 3D Laplacian underline our theoretical findings.

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Acknowledgments

The authors acknowledge support through the Austrian Science Fund (FWF) under Grant P27005 Optimal adaptivity for BEM and FEM-BEM coupling. MF and DP acknowledge the support of the FWF doctoral school Dissipation and Dispersion in Nonlinear PDEs, funded under Grant W1245. The research of TF is supported by the CONICYT project Preconditioned linear solvers for nonconforming boundary elements, funded under Grant FONDECYT 3150012.

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Authors and Affiliations

  1. Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstr. 8-10, 1040, Vienna, Austria

    Michael Feischl, Gregor Gantner, Alexander Haberl & Dirk Praetorius

  2. Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Avenida Vicuña Mackenna 4860, Santiago, Chile

    Thomas Führer

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  1. Michael Feischl
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  2. Gregor Gantner
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  4. Dirk Praetorius
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Correspondence to Dirk Praetorius.

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Feischl, M., Gantner, G., Haberl, A. et al. Adaptive boundary element methods for optimal convergence of point errors. Numer. Math. 132, 541–567 (2016). https://doi.org/10.1007/s00211-015-0727-4

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