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


First in this chapter we give a general framework of adaptive Petrov–Galerkin methods for the solution of operator equations in Banach spaces. This approach is made precise in the application to Symm’s integral equation. Then we present more general adaptive BEM. Here we use the residual error estimator and prove reliability and efficiency in 2D. Finally we analyze the hierarchical error estimator and demonstrate its applicability in two-level adaptive BEM for scalar and vector boundary value problems. Special emphasis is given to the 3D case for the weakly singular integral equation (Sect. 10.3) and for the hyper singular integral equation (Sect. 10.4). In Sect. 10.5 we present a two-level adaptive BEM for the weakly singular operator and the h-version on surface pieces. In Sect. 10.6 based on a two-level subspace decomposition for the p-version BEM we give hierarchical error estimators for the hypersingular integral operator on curves. Finally recent developments on the convergence of the adaptive BEM for the h-version are given in Sect. 10.7.


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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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