Wavenumber-explicit analysis for the Helmholtz h-BEM: error estimates and iteration counts for the Dirichlet problem

Jeffrey Galkowski, Eike Müller, Euan Spence

Research output: Contribution to journalArticle


We consider solving the exterior Dirichlet problem for the Helmholtz equation with the h-version of the boundary element method (BEM) using the standard second-kind combined-field integral equations. We prove a new, sharp bound on how the number of GMRES iterations must grow with the wavenumber k to have the error in the iterative solution bounded independently of k as k→∞ when the boundary of the obstacle is analytic and has strictly positive curvature. To our knowledge, this result is the first-ever sharp bound on how the number of GMRES iterations depends on the wavenumber for an integral equation used to solve a scattering problem. We also prove new bounds on how h must decrease with k to maintain k-independent quasi-optimality of the Galerkin solutions as k→∞ when the obstacle is nontrapping.
Original languageEnglish
Pages (from-to)329-357
Number of pages29
JournalNumerische Mathematik
Issue number2
Publication statusPublished - 1 Jun 2019


ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics

Cite this