## Abstract

We consider a wide variety of Helmholtz scattering problems including scattering by Dirichlet, Neumann, and penetrable obstacles. We consider a radial perfectly matched layer (PML) and show that for any fixed PML width and a steep-enough scaling angle, the PML solution is exponentially close, both in frequency and the tangent of the scaling angle, to the true scattering solution. Moreover, for a fixed scaling angle and large enough PML width, the PML solution is exponentially close to the true scattering solution in both frequency and the PML width. In fact, the exponential bound holds with rate of decay c(w tan θ-C)k, where w is the PML width and θ is the scaling angle. More generally, the results of the paper hold in the framework of black-box scattering under the assumption of an exponential bound on the norm of the cutoff resolvent, thus including problems with strong trapping. These are the first results on the exponential accuracy of PML at high-frequency with nontrivial scatterers.

Original language | English |
---|---|

Pages (from-to) | 3344-3394 |

Number of pages | 51 |

Journal | SIAM Journal on Mathematical Analysis |

Volume | 55 |

Issue number | 4 |

Early online date | 4 Aug 2023 |

DOIs | |

Publication status | Published - 31 Dec 2023 |

### Bibliographical note

Funding Information:*Received by the editors September 2, 2021; accepted for publication (in revised form) January 27, 2023; published electronically August 4, 2023. https://doi.org/10.1137/21M1443716 Funding: JG was supported by EPSRC grant EP/V001760/1, and DL and EAS were supported by EPSRC grant EP/R005591/1. \dagger Department of Mathematics, University College London, London, WC1H 0AY, UK (j.galkowski@ ucl.ac.uk). \ddagger CNRS and Institut de Math\e'matiques de Toulouse, UMR5219; Universit\e' de Toulouse, CNRS; UPS, F-31062 Toulouse Cedex 9, France (David.lafontaine@math.univ-toulouse.fr). \S Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY, UK (e.a.spence@ bath.ac.uk).

Publisher Copyright:

© 2023 Society for Industrial and Applied Mathematics.

## Keywords

- Helmholtz equation
- perfectly matched layer
- scattering

## ASJC Scopus subject areas

- Computational Mathematics
- Analysis
- Applied Mathematics