### Abstract

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 language | English |
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Pages (from-to) | 329-357 |

Number of pages | 29 |

Journal | Numerische Mathematik |

Volume | 142 |

Issue number | 2 |

DOIs | |

Publication status | Published - 1 Jun 2019 |

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### ASJC Scopus subject areas

- Computational Mathematics
- Applied Mathematics