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 $hp$-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. In particular, the result in (i) shows that the $hp$-FEM applied to this problem does not suffer from the pollution effect.

翻译：在过去十年中,[Melenk-Sauter,2010年]、[Melenk-Sauter,2011年]、[Esterhazy-Melenk-Melenk,2012年]和[Merenk-Parsania-Sauter,2013年]将高频Helmholtz溶液分解成“低”和“高”频率方程式的结果,对Helmholtz方程式的数值分析产生了重大影响。这些结果证明是,在Drichlet障碍或带有阻力边界条件的内部域内,常态的Helmholtz等方程式, 使用Helffer-Sj\'ostria-Sauterm-Sautermus溶解成高频 Helm-betz 方程式将问题分散在Sj\'o'ostrand-Zworski的黑盒框架内,从而覆盖了Helmztzt 问题,可变系数、不易性障碍和可穿的障碍。这些结果使我们得以证明, 应用新的频率-districal-lodial-comm-comm-comm-comm-rill rilli 的硬度的硬度结果, Exm-comm-rol Ex-rolational 问题在Exlation ex ex ex ex ex-stal Exl Exl 和 ex-st-stoltalmal Exl Exl ex-st-staltaltaldaldaldaldal 上, romaxil 问题在一种可变法 ex-stalticil (i)