对于边界具有盖瑞正则性的分段光滑亥姆霍兹问题，$hp$有限元法不存在污染效应
The $hp$-FEM does not suffer from the pollution effect for piecewise-smooth Helmholtz problems with Gevrey regularity at boundaries
AI总结：
研究亥姆霍兹散射问题，在特定条件下（如亥姆霍兹解算子在$k$上多项式有界等），证明当$p\geq 1+\varepsilon \log k$且$hk/p$足够小时，$hp$有限元法是拟最优的，不存在污染效应，推广了相关结果。
AI中文摘要：
我们考虑将$hp$有限元法应用于具有波数$k$的亥姆霍兹散射问题，并用完全匹配层截断。散射体由狄利克雷、诺伊曼和可穿透障碍物以及可变系数组合而成。假设亥姆霍兹解算子在$k$上多项式有界，所有系数分段光滑，所有边界面是盖瑞的，且所有限制在边界面上的系数及其法向导数都是盖瑞的，我们证明当$p\geq 1+\varepsilon \log k$且$hk/p$足够小时，$hp$有限元法是拟最优的，即不存在污染效应。此结果将[Bernkopf, Chaumont - Frelet, Melenk 2025]（针对分段解析系数和解析边界证明）以及[Galkowski, Lafontaine, Spence, Wunsch 2024]（针对在解析障碍物附近解析的光滑系数证明）的类似结果推广到了更大类的散射体。
英文摘要：
We consider the $hp$-FEM applied to the Helmholtz scattering problem with wavenumber $k$, truncated with a perfectly-matched layer. The scatterer consists of a combination of Dirichlet, Neumann, and penetrable obstacles together with variable coefficients. Provided that the Helmholtz solution operator is polynomially bounded in $k$, all coefficients are piecewise smooth, all boundary surfaces are Gevrey and all coefficients restricted to boundary surfaces are Gevrey together with all their normal derivatives, we show that the $hp$-FEM is quasioptimal when $p\geq 1+\varepsilon \log k$ and $hk/p$ is sufficiently small; i.e., the $hp$-FEM does not suffer from the pollution effect. This result generalises the analogous results in both [Bernkopf, Chaumont-Frelet, Melenk 2025] (proved for piecewise analytic coefficients and analytic boundaries) and [Galkowski, Lafontaine, Spence, Wunsch 2024] (proved for smooth coefficients that are analytic near analytic obstacles) to a much larger class of scatterers.
