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Calcolo

, Volume 54, Issue 1, pp 367–399 | Cite as

Optimal additive Schwarz preconditioning for hypersingular integral equations on locally refined triangulations

  • Michael Feischl
  • Thomas Führer
  • Dirk Praetorius
  • Ernst P. Stephan
Article

Abstract

For the non-preconditioned Galerkin matrix of the hypersingular integral operator, the condition number grows with the number of elements as well as the quotient of the maximal and the minimal mesh-size. Therefore, reliable and effective numerical computations, in particular on adaptively refined meshes, require the development of appropriate preconditioners. We propose and analyze a local multilevel preconditioner which is optimal in the sense that the condition number of the corresponding preconditioned system is independent of the number of elements, the local mesh-size, and the number of refinement levels. The theory covers closed boundaries as well as open screens in 2D and 3D. Numerical experiments underline the analytical results and compare the proposed preconditioner to other multilevel schemes as well as techniques based on operator preconditioning.

Keywords

Preconditioner Multilevel additive Schwarz Hypersingular integral equation 

Mathematics Subject Classification

65N30 65F08 65N38 

Notes

Acknowledgments

The research of the authors is supported by the Austrian Science Fund (FWF) through the research projects Adaptive boundary element method, funded under Grant P21732, and Optimal adaptivity of BEM and FEM-BEM coupling, funded under Grant P27005. In addition, the author TF is supported by the CONICYT project Preconditioned solvers for nonconforming boundary elements, funded under Grant FONDECYT 3150012.

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Copyright information

© Springer-Verlag Italia 2016

Authors and Affiliations

  • Michael Feischl
    • 1
  • Thomas Führer
    • 2
  • Dirk Praetorius
    • 3
  • Ernst P. Stephan
    • 4
  1. 1.School of Mathematics and StatisticsUniversity of New South WalesSydneyAustralia
  2. 2.Facultad de MatemáticasPontificia Universidad Católica de ChileSantiagoChile
  3. 3.Institute for Analysis and Scientific ComputingVienna University of TechnologyWienAustria
  4. 4.Institute for Applied MathematicsLeibniz University HannoverHannoverGermany

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