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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 steepenough 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 θC)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 blackbox 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 highfrequency with nontrivial scatterers.
Original language  English 

Pages (fromto)  33443394 
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, F31062 Toulouse Cedex 9, France ([email protected]). \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
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 1 Finished

At the interface between semiclassical analysis and numerical analysis of Wave propogation problems
Spence, E. (PI)
Engineering and Physical Sciences Research Council
1/10/17 → 30/09/23
Project: Research council