TY - JOUR

T1 - A sharp relative-error bound for the Helmholtz h-FEM at high frequency

AU - Lafontaine, D.

AU - Spence, E. A.

AU - Wunsch, J.

N1 - Funding Information:
We thank Théophile Chaumont-Frelet (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.
Publisher Copyright:
© 2021, The Author(s).

PY - 2021

Y1 - 2021

N2 - For the h-finite-element method (h-FEM) 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 k-explicit 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 fixed-order 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 relative-error bounds on the FEM solution for arbitrary fixed-order 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 plane-wave scattering problem, which we prove using semiclassical defect measures.

AB - For the h-finite-element method (h-FEM) 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 k-explicit 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 fixed-order 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 relative-error bounds on the FEM solution for arbitrary fixed-order 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 plane-wave scattering problem, which we prove using semiclassical defect measures.

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U2 - 10.1007/s00211-021-01253-0

DO - 10.1007/s00211-021-01253-0

M3 - Article

AN - SCOPUS:85120038182

JO - Numerische Mathematik

JF - Numerische Mathematik

SN - 0029-599X

ER -