Perfectly-matched-layer truncation is exponentially accurate at high frequency

TL;DR

This paper proves that perfectly-matched-layer (PML) truncation provides exponentially accurate solutions for high-frequency scattering problems, including complex obstacles, under certain conditions on PML width and scaling angle.

Contribution

It establishes the first exponential accuracy results for PML at high frequency for a broad class of scattering problems, including trapping scenarios.

Findings

PML solutions are exponentially close to true solutions with respect to frequency and PML parameters.

Exponential bounds depend on PML width, scaling angle, and frequency.

Results apply to a wide range of scattering problems, including those with trapping.

Abstract

We consider a wide variety of scattering problems including scattering by Dirichlet, Neumann, and penetrable obstacles. We consider a radial perfectly-matched layer (PML) and show that for any 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 $c(w\tan\theta -C) k$ where $w$ is the PML width and $\theta$ 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 non-trivial scatterers.