SIAM Review, Volume 65, Issue 3, Page 806-828, August 2023.
In $d$ dimensions, accurately approximating an arbitrary function oscillating with frequency $\lesssim k$ requires $\sim$$k^d$ degrees of freedom. A numerical method for solving the Helmholtz equation (with wavenumber $k$ and in $d$ dimensions) suffers from the pollution effect if, as $k→∞$, the total number of degrees of freedom needed to maintain accuracy grows faster than this natural threshold (i.e., faster than $k^d$ for domain-based formulations, such as finite element methods, and $k^{d-1}$ for boundary-based formulations, such as boundary element methods). It is well known that the $h$-version of the finite element method (FEM) (where accuracy is increased by decreasing the meshwidth $h$ and keeping the polynomial degree $p$ fixed) suffers from the pollution effect, and research over the last $\sim$30 years has resulted in a near-complete rigorous understanding of how quickly the number of degrees of freedom must grow with $k$ to maintain accuracy (and how this depends on both $p$ and properties of the scatterer). In contrast to the $h$-FEM, at least empirically, the $h$-version of the boundary element method (BEM) does not suffer from the pollution effect (recall that in the boundary element method the scattering problem is reformulated as an integral equation on the boundary of the scatterer, with this integral equation then solved numerically using a finite element--type approximation space). However, the current best results in the literature on how quickly the number of degrees of freedom for the $h$-BEM must grow with $k$ to maintain accuracy fall short of proving this. In this paper, we prove that the $h$-version of the Galerkin method applied to the standard second-kind boundary integral equations for solving the Helmholtz exterior Dirichlet problem does not suffer from the pollution effect when the obstacle is nontrapping (i.e., does not trap geometric-optic rays). While the proof of this result relies on information about the large-$k$ behavior of Helmholtz solution operators, we show in an appendix how the result can be proved using only Fourier series and asymptotics of Hankel and Bessel functions when the obstacle is a 2-d ball.

中文翻译：

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亥姆霍兹边界元法会受到污染影响吗？

《SIAM 评论》，第 65 卷，第 3 期，第 806-828 页，2023 年 8 月。
在 $d$ 维度中，精确逼近以频率 $\lesssim k$ 振荡的任意函数需要 $\sim$$k^d$ 自由度。求解亥姆霍兹方程（波数为 $k$，维度为 $d$）的数值方法会受到污染效应的影响，如果在 $k→∞$ 时，保持精度所需的自由度总数增长得快于这个值自然阈值（即，对于基于域的公式（例如有限元方法），比 $k^d$ 更快；对于基于边界的公式（例如边界元方法），比 $k^{d-1}$ 更快）。众所周知，有限元法 (FEM) 的 $h$ 版本（通过减小网格宽度 $h$ 并保持多项式次数 $p$ 固定来提高精度）受到污染效应的影响，过去 $\sim$30 年的研究已经对自由度的数量必须以多快的速度随 $k$ 增长才能保持准确性有了近乎完整的严格理解（以及这如何取决于 $p$ 和散射体）。与 $h$-FEM 相比，至少在经验上，边界元法 (BEM) 的 $h$-版本不会受到污染效应的影响（回想一下，在边界元法中，散射问题被重新表述为散射体边界上的积分方程，然后使用有限元（类型近似空间）对该积分方程进行数值求解。然而，目前文献中关于 $h$-BEM 的自由度数量必须随 $k$ 增长以保持精度的速度有多快的最佳结果还不足以证明这一点。在本文中，我们证明，应用于标准第二类边界积分方程来求解亥姆霍兹外狄利克雷问题的伽辽金方法的 $h$ 版本在障碍物非捕获（即不捕获几何-光线）。虽然这个结果的证明依赖于亥姆霍兹解算子的大$k$行为的信息，但我们在附录中展示了当障碍物是2时如何仅使用傅里叶级数以及汉克尔和贝塞尔函数的渐近来证明结果-d 球。