Perfectly-Matched-Layer Truncation is Exponentially Accurate at High Frequency

Abstract

We consider a wide variety of scattering problems including scattering by\nDirichlet, Neumann, and penetrable obstacles. We consider a radial\nperfectly-matched layer (PML) and show that for any PML width and a\nsteep-enough scaling angle, the PML solution is exponentially close, both in\nfrequency and the tangent of the scaling angle, to the true scattering\nsolution. Moreover, for a fixed scaling angle and large enough PML width, the\nPML solution is exponentially close to the true scattering solution in both\nfrequency and the PML width. In fact, the exponential bound holds with rate of\ndecay $c(w\\tan\\theta -C) k$ where $w$ is the PML width and $\\theta$ is the\nscaling angle. More generally, the results of the paper hold in the framework\nof black-box scattering under the assumption of an exponential bound on the\nnorm of the cutoff resolvent, thus including problems with strong trapping.\nThese are the first results on the exponential accuracy of PML at\nhigh-frequency with non-trivial scatterers.\n

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