Decompositions of high-frequency Helmholtz solutions via functional calculus, and application to the finite element method

Jeffrey Galkowski, David Lafontaine, Euan A. Spence, Jared Wunsch

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Over the last ten years, results from [48], [49], [22], and [47] 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 [33], this paper proves analogous decompositions for scattering problems fitting into the black-box scattering framework of Sj¨ostrandZworski [63], 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 hpf inite-element method (hp-FEM) applied to the variable-coefficient Helmholtz equation in the exterior of an analytic Dirichlet obstacle, where the coefficients are analytic in a neighbourhood of the obstacle, and (ii) the h-FEM applied to the Helmholtz penetrable-obstacle transmission problem. In particular, the result in (i) shows that the hp-FEM applied to this problem does not suffer from the pollution effect.
Original languageEnglish
Pages (from-to)3903-3958
Number of pages49
JournalSiam Journal on Mathematical Analysis
Issue number4
Early online date22 Aug 2023
Publication statusPublished - 31 Aug 2023

Bibliographical note

The authors thank Luis Escauriaza (Universidad del Pas Vasco/Euskal Herriko Unibertsitatea)
for useful discussions related to the paper [21]. We also thank the referees and the associate editor
for their constructive comments that improved the organisation of the paper. JG acknowledges
support from EPSRC grant EP/V001760/1. DL and EAS acknowledge support from EPSRC grant
EP/1025995/1. JW was partly supported by Simons Foundation grant 631302 and by NSF grant


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