eprintid: 10204409
rev_number: 10
eprint_status: archive
userid: 699
dir: disk0/10/20/44/09
datestamp: 2025-02-07 14:10:53
lastmod: 2025-02-07 14:10:53
status_changed: 2025-02-07 14:10:53
type: article
metadata_visibility: show
sword_depositor: 699
creators_name: Galkowski, J
creators_name: Spence, EA
title: Sharp Preasymptotic Error Bounds for the Helmholtz h-FEM
ispublished: pub
divisions: UCL
divisions: B04
divisions: C06
divisions: F59
keywords: Helmholtz, FEM, pollution effect, high order, perfectly matched layer, preasymptotic, elliptic projection
note: This version is the author-accepted manuscript. For information on re-use, please refer to the publisher’s terms and conditions.
abstract: In the analysis of the h-version of the finite-element method (FEM), with fixed polynomial degree p, applied to the Helmholtz equation with wavenumber k >> 1, the asymptotic regime is when (hk)ᴾC sol is sufficiently small and the sequence of Galerkin solutions are quasioptimal; here Csol is the L² L² norm of the Helmholtz solution operator, with Csol ~ k for nontrapping problems. In the preasymptotic regime, one expects that if (hk)²ᵖC sol is sufficiently small, then (for physical data) the relative error of the Galerkin solution is controllably small. In this paper, we prove the natural error bounds in the preasymptotic regime for the variable-coefficient Helmholtz equation in the exterior of a Dirichlet, or Neumann, or penetrable obstacle (or combinations of these) and with the radiation condition either realized exactly using the Dirichlet-to-Neumann map on the boundary of a ball or approximated either by a radial perfectly matched layer (PML) or an impedance boundary condition. Previously, such bounds for p > 1 were only available for Dirichlet obstacles with the radiation condition approximated by an impedance boundary condition. Our result is obtained via a novel generalization of the ``elliptic-projection"" argument (the argument used to obtain the result for p = 1), which can be applied to a wide variety of abstract Helmholtz-type problems.
date: 2025-02
date_type: published
publisher: Society for Industrial & Applied Mathematics (SIAM)
official_url: https://doi.org/10.1137/23M1546178
oa_status: green
full_text_type: other
language: eng
primo: open
primo_central: open_green
verified: verified_manual
elements_id: 2357463
doi: 10.1137/23M1546178
lyricists_name: Galkowski, Jeffrey Eric
lyricists_id: JGALK87
actors_name: Galkowski, Jeffrey Eric
actors_id: JGALK87
actors_role: owner
full_text_status: public
publication: SIAM Journal on Numerical Analysis
volume: 63
number: 1
pagerange: 1-22
issn: 1095-7170
citation:        Galkowski, J;    Spence, EA;      (2025)    Sharp Preasymptotic Error Bounds for the Helmholtz h-FEM.                   SIAM Journal on Numerical Analysis , 63  (1)   pp. 1-22.    10.1137/23M1546178 <https://doi.org/10.1137/23M1546178>.       Green open access   
 
document_url: https://discovery.ucl.ac.uk/id/eprint/10204409/1/preasymptotic_final.pdf