2026/07/17 by Jeffrey Galkowski, Mostafa Meliani, Euan A. Spence
#math.NA #cs.NA
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≥ 1+ε 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.