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. 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.

中文翻译：

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基于函数演算的高频亥姆霍兹解的分解及其在有限元方法中的应用

在过去十年中，[Melenk-Sauter，2010]，[Melenk-Sauter，2011]，[Esterhazy-Melenk，2012]和[Melenk-Parsania-Sauter，2013]的结果将高频Helmholtz解决方案分解为“ “低频”和“高频”分量对亥姆霍兹方程的数值分析产生了很大的影响。对于Dirichlet障碍物外部或具有阻抗边界条件的内部区域中的常数系数Helmholtz方程，已经证明了这些结果。使用Helffer-Sj“ ostrand函数演算，本文证明了适合Sj” ostrand-Zworski黑盒散射框架的散射问题的类似分解，从而涵盖了变系数，不可穿透的障碍和可穿透的障碍的亥姆霍兹问题一次全部。特别是，