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Wavelet Boundary Element Methods: Adaptivity and Goal-Oriented Error Estimation

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Advanced Finite Element Methods with Applications (FEM 2017)

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 128))

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Abstract

This article is dedicated to the adaptive wavelet boundary element method. It computes an approximation to the unknown solution of the boundary integral equation under consideration with a rate \(N_{\text{dof}}^{-s}\), whenever the solution can be approximated with this rate in the setting determined by the underlying wavelet basis. The computational cost scale linearly in the number N dof of degrees of freedom. Goal-oriented error estimation for evaluating linear output functionals of the solution is also considered. An algorithm is proposed that approximately evaluates a linear output functional with a rate \(N_{\text{dof}}^{-(s+t)}\), whenever the primal solution can be approximated with a rate \(N_{\text{dof}}^{-s}\) and the dual solution can be approximated with a rate \(N_{\text{dof}}^{-t}\), while the cost still scale linearly in N dof. Numerical results for an acoustic scattering problem and for the point evaluation of the potential in case of the Laplace equation are reported to validate and quantify the approach.

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Notes

  1. 1.

    In accordance with [23], one might also consider a compact perturbation of an elliptic, symmetric, and continuous boundary integral operator.

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Acknowledgements

This research has been supported by the Swiss National Science Foundation (SNSF) through the DACH-project “BIOTOP: Adaptive Wavelet and Frame Techniques for Acoustic BEM”.

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Authors and Affiliations

  1. Departement Mathematik und Informatik, Universität Basel, Basel, Switzerland

    Helmut Harbrecht

  2. Universitätsspital Basel, Basel, Switzerland

    Manuela Moor

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  1. Helmut Harbrecht
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  2. Manuela Moor
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Correspondence to Helmut Harbrecht .

Editor information

Editors and Affiliations

  1. Institut für Mathematik & Computergestützte Simulation, Universität der Bundeswehr München, Neubiberg, Germany

    Thomas Apel

  2. Institute for Computational Mathematics, Johannes Kepler University Linz, Linz, Austria

    Ulrich Langer

  3. Fakultät für Mathematik, TU Chemnitz, Chemnitz, Germany

    Arnd Meyer

  4. Institut für Angewandte Mathematik, Technische Universität Graz, Graz, Austria

    Olaf Steinbach

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Harbrecht, H., Moor, M. (2019). Wavelet Boundary Element Methods: Adaptivity and Goal-Oriented Error Estimation. In: Apel, T., Langer, U., Meyer, A., Steinbach, O. (eds) Advanced Finite Element Methods with Applications. FEM 2017. Lecture Notes in Computational Science and Engineering, vol 128. Springer, Cham. https://doi.org/10.1007/978-3-030-14244-5_8

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