The $hp$-FEM does not suffer from the pollution effect for piecewise-smooth Helmholtz problems with Gevrey regularity at boundaries

Abstract

We consider the $hp$-FEM applied to the Helmholtz scattering problem with wavenumber $k$, truncated with a perfectly-matched layer. The scatterer consists of a combination of Dirichlet, Neumann, and penetrable obstacles together with variable coefficients. Provided that the Helmholtz solution operator is polynomially bounded in $k$, all coefficients are piecewise smooth, all boundary surfaces are Gevrey and all coefficients restricted to boundary surfaces are Gevrey together with all their normal derivatives, we show that the $hp$-FEM is quasioptimal when $p\ge 1+\epsilon \log k$ and $hk/p$ is sufficiently small; i.e., the $hp$-FEM does not suffer from the pollution effect. This result generalises the analogous results in both [Bernkopf, Chaumont-Frelet, Melenk 2025] (proved for piecewise analytic coefficients and analytic boundaries) and [Galkowski, Lafontaine, Spence, Wunsch 2024] (proved for smooth coefficients that are analytic near analytic obstacles) to a much larger class of scatterers.