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Abstract
For the hfiniteelement method (hFEM) applied to the Helmholtz equation, the question of how quickly the meshwidth h must decrease with the frequency k to maintain accuracy as k increases has been studied since the mid 80’s. Nevertheless, there still do not exist in the literature any kexplicit bounds on the relative error of the FEM solution (the measure of the FEM error most often used in practical applications), apart from in one dimension. The main result of this paper is the sharp result that, for the lowest fixedorder conforming FEM (with polynomial degree, p, equal to one), the condition “h2k3 sufficiently small" is sufficient for the relative error of the FEM solution in 2 or 3 dimensions to be controllably small (independent of k) for scattering of a plane wave by a nontrapping obstacle and/or a nontrapping inhomogeneous medium. We also prove relativeerror bounds on the FEM solution for arbitrary fixedorder methods applied to scattering by a nontrapping obstacle, but these bounds are not sharp for p≥2. A key ingredient in our proofs is a result describing the oscillatory behaviour of the solution of the planewave scattering problem, which we prove using semiclassical defect measures.
Original language  English 

Pages (fromto)  137178 
Number of pages  42 
Journal  Numerische Mathematik 
Volume  150 
Issue number  1 
Early online date  27 Nov 2021 
DOIs  
Publication status  Published  Jan 2022 
Bibliographical note
Funding Information:We thank Théophile ChaumontFrelet (INRIA, Nice), Ivan Graham (University of Bath), and particularly Owen Pembery (University of Bath) for useful discussions. We thank the referees for numerous useful comments and suggestions. DL and EAS acknowledge support from EPSRC Grant EP/1025995/1. JW was partly supported by Simons Foundation Grant 631302.
ASJC Scopus subject areas
 Computational Mathematics
 Applied Mathematics
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Dive into the research topics of 'A sharp relativeerror bound for the Helmholtz hFEM at high frequency'. Together they form a unique fingerprint.Projects
 1 Finished

At the interface between semiclassical analysis and numerical analysis of Wave propogation problems
Spence, E. (PI)
Engineering and Physical Sciences Research Council
1/10/17 → 30/09/23
Project: Research council