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We consider approximation of the variable-coefficient Helmholtz equation inthe exterior of a Dirichlet obstacle using perfectly-matched-layer (PML)truncation; it is well known that this approximation is exponentially accuratein the PML width and the scaling angle, and the approximation was recentlyproved 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 thepollution 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 (withconstant independent of $k$), under the following two conditions (i) thesolution 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 theobstacle is analytic and the coefficients are analytic in a neighbourhood ofthe 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 theoriginal Helmholtz solution recently proved in [Galkowski, Lafontaine, Spence,Wunsch, 2022]. The decomposition is obtained using tools from semiclassicalanalysis (i.e., the PDE techniques specifically designed for studying Helmholtzproblems with large $k$).

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