@article{discovery10185834, volume = {22}, publisher = {International Press of Boston, Inc.}, year = {2024}, title = {The hp-FEM applied to the Helmholtz equation with PML truncation does not suffer from the pollution effect}, number = {7}, journal = {Communications in Mathematical Sciences}, pages = {1761--1816}, note = {This version is the author accepted manuscript. For information on re-use, please refer to the publisher's terms and conditions.}, author = {Galkowski, J and Lafontaine, D and Spence, E A and Wunsch, J}, url = {https://dx.doi.org/10.4310/CMS.240918021620}, 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 [28]. We show that the hp-FEM applied to this problem does not suffer from the pollution effect, in that there exist C1, C2 {\ensuremath{>}} 0 such that if hk/p {$\leq$} C1 and p {$\ge$} 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).}, keywords = {Helmholtz equation, high frequency, perfectly-matched layer, pollution effect, finite element method, error estimate, semiclassical analysis} }