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

02/25/2021
by   David Lafontaine, et al.
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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-Sjöstrand functional calculus, this paper proves analogous decompositions for scattering problems fitting into the black-box scattering framework of Sjöstrand-Zworski, thus covering Helmholtz problems with variable coefficients, impenetrable obstacles, and penetrable obstacles all at once. In particular, these results allow us to prove new frequency-explicit convergence results for (i) the hp-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 h-finite-element method applied to the Helmholtz penetrable-obstacle transmission problem.

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