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Exponential Convergence of hp-BEM

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Advanced Boundary Element Methods

Part of the book series: Springer Series in Computational Mathematics ((SSCM,volume 52))

Abstract

The first section of this chapter collects results from [240] which gives a further contribution to the analysis of the hp-version of the boundary element method (BEM) by presenting a more general result for Dirichlet and Neumann problems than [21] allowing the use of a general geometric mesh refinement on the polygonal boundary Γ. Here as in [240] we prove the exponential convergence of the hp- version of the boundary element method by exploiting only features of the solutions of the boundary integral equations. The key result in this approach is an asymptotic expansion of the solution of the integral equations in singularity functions reflecting the singular behaviour of the solutions near corners of Γ. With such expansions we show that the solutions of the integral equations belong to countably normed spaces. Therefore these solutions can be approximated exponentially fast in the energy norm via the hp- Galerkin solutions of those integral equations. This result is not restricted to integral equations which stem from boundary value problems for the Laplacian but applies to Helmholtz problems as well. Further applications are 2D crack problems in linear elasticity. For numerical experiments with hp-version (BEM) see [165, 340].

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Authors and Affiliations

  1. Fakultät für Luft- und Raumfahrttechnik, Universität der Bundeswehr München, Neubiberg/München, Germany

    Joachim Gwinner

  2. Institut für Angewandte Mathematik, Leibniz Universität Hannover, Hannover, Germany

    Ernst Peter Stephan

Authors
  1. Joachim Gwinner
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  2. Ernst Peter Stephan
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Gwinner, J., Stephan, E.P. (2018). Exponential Convergence of hp-BEM. In: Advanced Boundary Element Methods. Springer Series in Computational Mathematics, vol 52. Springer, Cham. https://doi.org/10.1007/978-3-319-92001-6_8

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