%0 Journal Article %A Galkowski, J %A Lafontaine, D %A Spence, E A %A Wunsch, J %D 2024 %F discovery:10185834 %I International Press of Boston, Inc. %J Communications in Mathematical Sciences %K Helmholtz equation, high frequency, perfectly-matched layer, pollution effect, finite element method, error estimate, semiclassical analysis %N 7 %P 1761-1816 %T The hp-FEM applied to the Helmholtz equation with PML truncation does not suffer from the pollution effect %U https://discovery.ucl.ac.uk/id/eprint/10185834/ %V 22 %X 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 [28]. We show that the hp-FEM applied to this problem does not suffer from the pollution effect, in that there exist C1, C2 > 0 such that if hk/p ≤ C1 and p ≥ C2 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 [41]), 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 [29]. The decomposition is obtained using tools from semiclassical analysis (i.e., the PDE techniques specifically designed for studying Helmholtz problems with large k). %Z This version is the author accepted manuscript. For information on re-use, please refer to the publisher’s terms and conditions.