We consider approximation of the variable-coefficient Helmholtz equation in the exterior of a Dirichlet obstacle using perfectly-matched-layer (PML) truncation; it is well known that this approximation is exponentially accurate in the PML width and the scaling angle, and the approximation was recently proved to be exponentially accurate in the wavenumber $k$ in [Galkowski, Lafontaine, Spence, 2021]. We show that the $hp$-FEM applied to this problem does not suffer from the pollution effect, in that there exist $C_1,C_2>0$ such that if $hk/p\leq C_1$ and $p \geq C_2 \log k$ then the Galerkin solutions are quasioptimal (with constant independent of $k$), under the following two conditions (i) the solution operator of the original Helmholtz problem is polynomially bounded in $k$ (which occurs for "most" $k$ by [Lafontaine, Spence, Wunsch, 2021]), and (ii) either there is no obstacle and the coefficients are smooth or the obstacle is analytic and the coefficients are analytic in a neighbourhood of the obstacle and smooth elsewhere. This $hp$-FEM result is obtained via a decomposition of the PML solution into "high-" and "low-frequency" components, analogous to the decomposition for the original Helmholtz solution recently proved in [Galkowski, Lafontaine, Spence, Wunsch, 2022]. The decomposition is obtained using tools from semiclassical analysis (i.e., the PDE techniques specifically designed for studying Helmholtz problems with large $k$).

翻译：本文研究了在无界障碍物外部带变系数的亥姆霍兹方程采用完美匹配层（PML）截断的逼近问题。已知该逼近在PML宽度和缩放角度上具有指数精度，且最近[Galkowski, Lafontaine, Spence, 2021]证明了其在波数$k$上同样具有指数精度。我们证明应用于该问题的$hp$-有限元方法不受污染效应影响：存在常数$C_1,C_2>0$，使得当满足以下两个条件时，Galerkin解是拟最优的（常数与$k$无关）：(i) 原始亥姆霍兹问题的解算子关于$k$具有多项式有界性（根据[Lafontaine, Spence, Wunsch, 2021]，这对"大多数"$k$成立）；(ii) 要么不存在障碍物且系数光滑，要么障碍物解析且系数在其邻域内解析而在其他区域光滑。该$hp$-有限元结果通过将PML解分解为"高频"和"低频"分量得到，类似于近期[Galkowski, Lafontaine, Spence, Wunsch, 2022]证明的原始亥姆霍兹解分解。该分解使用半经典分析工具（即专门设计用于研究大$k$亥姆霍兹问题的偏微分方程技术）获得。