On efficient preconditioners for iterative solution of a Galerkin boundary element equation for the three-dimensional exterior Helmholtz problem

Paul Harris, K. Chen

Research output: Contribution to journalArticle

Abstract

The paper presents a Galerkin numerical method for solving the hyper-singular boundary integral equations for the exterior Helmholtz problem in three dimensions with a Neumann's boundary condition. Previous work in the topic has often dealt with the collocation method with a piecewise constant approximation because high order collocation and Galerkin methods are not available due to the presence of a hypersingular integral operator. This paper proposes a high order Galerkin method by using singularity subtraction technique to reduce the hyper-singular operator to a weakly singular one. Moreover, we show here how to extend the previous work (J. Appl. Numer. Math. 36 (4) (2001) 475–489) on sparse preconditioners to the Galerkin case leading to fast convergence of two iterative solvers: the conjugate gradient normal method and the generalised minimal residual method. A comparison with the collocation method is also presented for the Helmholtz problem with several wavenumbers.
Original languageEnglish
Pages (from-to)303-318
Number of pages16
JournalJournal of Computational and Applied Mathematics
Volume156
Issue number2
Publication statusPublished - Jul 2003

Keywords

  • Exterior Helmholtz
  • Boundary integral equation
  • Burton–Miller
  • Hyper-singular operators
  • Galerkin
  • Preconditioners
  • CGN
  • GMRES

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