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

TL;DR

This work develops sharp preasymptotic error bounds for the $h$-FEM applied to the Helmholtz equation in exterior domains with variable coefficients and fixed polynomial degree. By extending the elliptic-projection argument through a regularizing operator, the authors prove bounds that hold in the preasymptotic regime whenever $(hk)^{2p} C_{ m sol}$ is small, for truncations realized by the exact DtN map, radial PML, or impedance boundary conditions. The contributions include a general abstract theorem and its verification for PML and exact truncations, yielding $H^1_k$ and $L^2$ error bounds and a relative $H^1_k$ error bound that scale with $(hk)^p$ under suitable data regularity. These results inform mesh design and pollution control for high-frequency Helmholtz simulations in exterior scattering problems and extend known $p=1$ results to arbitrary $p$ under standard elliptic-regularity assumptions.

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.

Theorems & Definitions (33)

• Theorem 1.1: Informal statement of the main result
• Remark 1.3
• Example 1.4
• Theorem 1.5: Abstract generalisation of the elliptic-projection argument
• Theorem 1.7: Bound on $\eta({\@fontswitch{}{\mathcal{}} H}_h)$
• Example 1.8
• Lemma 2.1
• Remark 2.2: Relation with the original elliptic-projection argument
• Proof 1: Proof of Lemma \ref{['lem:abs1']}
• Remark 2.3: The splitting \ref{['eq:niceFormula']}