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Advanced BEM for BVPs in Polygonal/Polyhedral Domains: h- and p-Versions

  • Joachim Gwinner
  • Ernst Peter Stephan
Chapter
Part of the Springer Series in Computational Mathematics book series (SSCM, volume 52)

Abstract

This chapter presents, h−, p −BEM on graded meshes and hp −BEM on quasiuniform meshes for the numerical treatment of boundary value problems in polygonal and polyhedral domains. For ease of presentation we also introduce here the hp −version on geometrically graded meshes (for details and proofs see Chap.  8). For the solutions of Dirichlet and Neumann problems we present decompositions into a sum of special singularity terms (describing their edge and corner behaviors) and in regular parts (see Theorem 7.3, Theorem 7.12 for two-dimensions and Theorem 7.7, Theorem 7.16 for three dimensions). These regularity results by von Petersdorff, Stephan [425] are based on the seminal works of Dauge [141] and Kondratiev [270]. Chapter 7 is organized as follows: The results for the single layer integral equation covering the Dirichlet problem are presented in Sect. 7.1 ; those for the hypersingular integral equation covering the Neumann problem in Sect. 7.2. Then in Sect. 7.3 the proofs for the results for the integral equations on curves are given, whereas in Sect. 7.4 the results for the integral equations on surfaces. We present approximation results for solutions of the integral equations on graded meshes in 2D and 3D from the PhD thesis by von Petersdorff [423], see also [426]. Also in detail we investigate the hp −version of BEM on quasi uniform meshes on polygons based on the paper by Suri and Stephan [405]. For the p-version BEM with quasi uniform meshes on polyhedra we refer to [51, 52, 374].

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Joachim Gwinner
    • 1
  • Ernst Peter Stephan
    • 2
  1. 1.Fakultät für Luft- und RaumfahrttechnikUniversität der Bundeswehr MünchenNeubiberg/MünchenGermany
  2. 2.Institut für Angewandte MathematikLeibniz Universität HannoverHannoverGermany

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