Decompositions of high-frequency Helmholtz solutions via functional calculus, and application to the finite element method
Abstract
Over the last ten years, results from [Melenk-Sauter, 2010], [Melenk-Sauter, 2011], [Esterhazy-Melenk, 2012], and [Melenk-Parsania-Sauter, 2013] decomposing high-frequency Helmholtz solutions into "low"- and "high"-frequency components have had a large impact in the numerical analysis of the Helmholtz equation. These results have been proved for the constant-coefficient Helmholtz equation in either the exterior of a Dirichlet obstacle or an interior domain with an impedance boundary condition. Using the Helffer-Sj\"ostrand functional calculus, this paper proves analogous decompositions for scattering problems fitting into the black-box scattering framework of Sj\"ostrand-Zworski, thus covering Helmholtz problems with variable coefficients, impenetrable obstacles, and penetrable obstacles all at once. These results allow us to prove new frequency-explicit convergence results for (i) the -finite-element method applied to the variable coefficient Helmholtz equation in the exterior of a Dirichlet obstacle, when the obstacle and coefficients are analytic, and (ii) the -finite-element method applied to the Helmholtz penetrable-obstacle transmission problem. In particular, the result in (i) shows that the -FEM applied to this problem does not suffer from the pollution effect.
Keywords
Cite
@article{arxiv.2102.13081,
title = {Decompositions of high-frequency Helmholtz solutions via functional calculus, and application to the finite element method},
author = {Jeffrey Galkowski and David Lafontaine and Euan A. Spence and Jared Wunsch},
journal= {arXiv preprint arXiv:2102.13081},
year = {2022}
}