SIAM Journal on Scientific Computing, Volume 43, Issue 2, Page A1027-A1061, January 2021.
In this paper, we design a truly exact perfect absorbing layer (PAL) for domain truncation of the two-dimensional Helmholtz equation in an unbounded domain with bounded scatterers. This technique is based on a complex compression coordinate transformation in polar coordinates and a judicious substitution of the unknown field in the artificial layer. Compared with the widely used perfectly matched layer (PML) methods, the distinctive features of PAL lie in that (i) it is truly exact in the sense that the PAL-solution is identical to the original solution in the bounded domain reduced by the truncation layer; (ii) with the substitution, the PAL-equation is free of singular coefficients and the substituted unknown field is essentially nonoscillatory in the layer; and (iii) the construction is valid for general star-shaped domain truncation. By formulating the variational formulation in Cartesian coordinates, the implementation of this technique using standard spectral-element or finite-element methods can be made easy as a usual coding practice. We provide ample numerical examples to demonstrate that this method is highly accurate and robust for very high wavenumbers and thin layers. It outperforms the classical PML and the recently advocated PML using unbounded absorbing functions. Moreover, it can fix some flaws of the PML approach.

中文翻译：

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一个真正精确的完美吸收层，用于时谐声波散射问题

SIAM科学计算杂志，第43卷，第2期，第A1027-A1061页，2021年1月。
在本文中，我们设计了一个真正精确的完美吸收层（PAL），用于带边界散射体的无界域中二维Helmholtz方程的域截断。该技术基于极坐标中的复杂压缩坐标变换以及对人工层中未知场的明智替换。与广泛使用的完全匹配层（PML）方法相比，PAL的独特之处在于：（i）在截断减少的有界域中，PAL解与原始解完全相同，这是真正准确的。层; （ii）通过替换，PAL方程不包含奇异系数，并且替换的未知字段在该层中基本上是非振荡的；（iii）该构造适用于一般的星形域截断。通过用笛卡尔坐标表示变化公式，可以很容易地使用标准的频谱元素或有限元素方法来实现该技术，这是通常的编码做法。我们提供了足够的数值示例，以证明该方法对于非常高的波数和薄层来说是高度准确且稳健的。它使用无界吸收功能胜过经典PML和最近提倡的PML。而且，它可以修复PML方法的一些缺陷。它使用无界吸收功能胜过经典PML和最近提倡的PML。而且，它可以修复PML方法的一些缺陷。它使用无界吸收功能胜过经典PML和最近提倡的PML。而且，它可以修复PML方法的一些缺陷。