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