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.
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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.
| Type: | Article |
|---|---|
| Title: | Sharp Preasymptotic Error Bounds for the Helmholtz h-FEM |
| Open access status: | An open access version is available from UCL Discovery |
| DOI: | 10.1137/23M1546178 |
| Publisher version: | https://doi.org/10.1137/23M1546178 |
| Language: | English |
| Additional information: | This version is the author-accepted manuscript. For information on re-use, please refer to the publisher’s terms and conditions. |
| Keywords: | Helmholtz, FEM, pollution effect, high order, perfectly matched layer, preasymptotic, elliptic projection |
| UCL classification: | UCL UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics |
| URI: | https://discovery.ucl.ac.uk/id/eprint/10204409 |
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