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

TL;DR

This paper establishes a sharp, explicit bound on the mesh size relative to frequency for the $h$-FEM applied to the Helmholtz equation, ensuring controllably small relative error in high-frequency scattering problems.

Contribution

It provides the first sharp, $k$-explicit relative-error bounds for the lowest-order $h$-FEM in 2D and 3D Helmholtz scattering, using semiclassical analysis.

Findings

For $p=1$, $h^2 k^3$ small ensures small relative error.

Bounds are extended to higher-order methods, but are not sharp for $p extgreater 1$.

Oscillatory behavior of solutions characterized using semiclassical defect measures.

Abstract

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 "$h^2 k^3$ 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 $p\geq 2$. 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.