论文标题
Helmholtz边界元素方法是否患有污染效果？
Does the Helmholtz boundary element method suffer from the pollution effect?
论文作者
论文摘要
在$ d $尺寸中，准确地近似使用频率$ \ lyssim k $振荡的任意函数需要$ \ sim k^d $自由度。如果$ k \ to \ k \ to \ iftty $，维持准确性的自由度比这种自然阈值更快（即比$ k^d $更快的difient e元素），则是一种求解Helmholtz方程（带有波形$ k $和$ d $尺寸）的数值方法具有污染效果用于基于边界的公式，例如边界元素方法）。 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. 与$ h $ -fem相反，至少从经验上讲，边界元素方法（BEM）的$ h $转换确实具有污染效果（请回想一下，在边界元素方法中，在边界元素方法中，散射问题在散点器的边界上被重新定程为一个积分方程，然后使用该积分方程式使用一个数值 - 使用数字 - 依次 - 求解数值。但是，文献中目前的最佳结果是关于$ h $ bem的自由度的速度必须在$ k $的情况下增长，以保持准确度未能证明这一点。 在本文中，我们证明，Galerkin方法的$ h $ version应用于标准的第二种边界积分方程，以解决Helmholtz外部Dirichlet问题不会在障碍物未捕获时遭受污染效应（即，不会陷入陷阱的几何光线射线）。
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 $k\to\infty$, 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. In contrast to the $h$-FEM, at least empirically, the $h$-version of the boundary element method (BEM) does $\textit{not}$ 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).
