The $hp$-FEM applied to the Helmholtz equation with PML truncation does not suffer from the pollution effect

Abstract

We consider approximation of the variable-coefficient Helmholtz equation in the exterior of a Dirichlet obstacle using perfectly-matched-layer (PML) truncation; it is well known that this approximation is exponentially accurate in the PML width and the scaling angle, and the approximation was recently proved to be exponentially accurate in the wavenumber $k$ in [Galkowski, Lafontaine, Spence, 2021]. We show that the $hp$-FEM applied to this problem does not suffer from the pollution effect, in that there exist $C_1,C_2>0$ such that if $hk/p\leq C_1$ and $p \geq C_2 \log k$ then the Galerkin solutions are quasioptimal (with constant independent of $k$), under the following two conditions (i) the solution operator of the original Helmholtz problem is polynomially bounded in $k$ (which occurs for "most" $k$ by [Lafontaine, Spence, Wunsch, 2021]), and (ii) either there is no obstacle and the coefficients are smooth or the obstacle is analytic and the coefficients are analytic in a neighbourhood of the obstacle and smooth elsewhere. This $hp$-FEM result is obtained via a decomposition of the PML solution into "high-" and "low-frequency" components, analogous to the decomposition for the original Helmholtz solution recently proved in [Galkowski, Lafontaine, Spence, Wunsch, 2022]. The decomposition is obtained using tools from semiclassical analysis (i.e., the PDE techniques specifically designed for studying Helmholtz problems with large $k$).