eprintid: 10185834 rev_number: 15 eprint_status: archive userid: 699 dir: disk0/10/18/58/34 datestamp: 2024-01-19 08:14:40 lastmod: 2024-12-03 16:04:04 status_changed: 2024-06-07 15:03:17 type: article metadata_visibility: show sword_depositor: 699 creators_name: Galkowski, J creators_name: Lafontaine, D creators_name: Spence, E A creators_name: Wunsch, J title: The hp-FEM applied to the Helmholtz equation with PML truncation does not suffer from the pollution effect ispublished: pub divisions: UCL divisions: B04 divisions: C06 divisions: F59 keywords: Helmholtz equation, high frequency, perfectly-matched layer, pollution effect, finite element method, error estimate, semiclassical analysis note: This version is the author accepted manuscript. For information on re-use, please refer to the publisher’s terms and conditions. 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 > 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). date: 2024 date_type: published publisher: International Press of Boston, Inc. official_url: https://dx.doi.org/10.4310/CMS.240918021620 oa_status: green full_text_type: other language: eng primo: open primo_central: open_green verified: verified_manual elements_id: 2140479 doi: 10.4310/CMS.240918021620 lyricists_name: Galkowski, Jeffrey Eric lyricists_id: JGALK87 actors_name: Galkowski, Jeffrey Eric actors_id: JGALK87 actors_role: owner funding_acknowledgements: EP/V001760/1 [Engineering and Physical Sciences Research Council]; EP/V051636/1 [Engineering and Physical Sciences Research Council]; EP/1025995/1. [Engineering and Physical Sciences Research Council]; DMS–2054424, [National Science Foundation]; 631302 [Simons Foundation] full_text_status: public publication: Communications in Mathematical Sciences volume: 22 number: 7 pagerange: 1761-1816 citation: Galkowski, J; Lafontaine, D; Spence, E A; Wunsch, J; (2024) The hp-FEM applied to the Helmholtz equation with PML truncation does not suffer from the pollution effect. Communications in Mathematical Sciences , 22 (7) pp. 1761-1816. 10.4310/CMS.240918021620 <https://doi.org/10.4310/CMS.240918021620>. Green open access document_url: https://discovery.ucl.ac.uk/id/eprint/10185834/1/GLSW1.pdf