Sharp preasymptotic error bounds for the Helmholtz $h$-FEM

TL;DR

This paper establishes sharp preasymptotic error bounds for the $h$-FEM applied to the Helmholtz equation with high wavenumber, extending previous results to variable coefficients and various boundary conditions.

Contribution

It provides the first rigorous preasymptotic error bounds for $p>1$ in the Helmholtz $h$-FEM, using a novel elliptic-projection approach applicable to diverse Helmholtz problems.

Findings

Proved error bounds in the preasymptotic regime for variable-coefficient Helmholtz problems.

Extended error analysis to boundary conditions including PML and impedance.

Generalized the elliptic-projection argument for broader applicability.

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 $k\gg 1$, the $\textit{asymptotic regime}$ is when $(hk)^p C_{\rm sol}$ is sufficiently small and the sequence of Galerkin solutions are quasioptimal; here $C_{\rm sol}$ is the $L^2 \to L^2$ norm of the Helmholtz solution operator, with $C_{\rm sol} \sim k$ for nontrapping problems. In the $\textit{preasymptotic regime}$, one expects that if $(hk)^{2p}C_{\rm sol}$ 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 $\textit{either}$ realised exactly using the Dirichlet-to-Neumann map on the boundary of a ball $\textit{or}$ 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 generalisation 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.