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

TL;DR

This paper proves that the $hp$-finite element method applied to the Helmholtz equation with PML truncation avoids the pollution effect, achieving quasioptimal solutions under certain conditions related to mesh size, polynomial degree, and problem regularity.

Contribution

It demonstrates that the $hp$-FEM can be quasioptimal for the Helmholtz problem with PML, without suffering from pollution, under specific mesh and polynomial degree conditions.

Findings

$hp$-FEM solutions are quasioptimal if $hk/p o 0$ and $p o \log k$.

The method avoids pollution effect under conditions on mesh size and polynomial degree.

Decomposition of PML solutions into frequency components is key to the analysis.

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 $k$).