We introduce a new class of absorbing boundary conditions (ABCs) for the Helmholtz equation. The proposed ABCs can be derived from a certain simple class of perfectly matched layers using $L$ discrete layers and using the $Q_N$ Lagrange finite element in conjunction with the $N$-point Gauss-Legendre quadrature reduced integration rule. The proposed ABCs are classified by a tuple $(L,N)$, and achieve reflection error of order $O(R^{2LN})$ for some $R<1$. The new ABCs generalise the perfectly matched discrete layers proposed by Guddati and Lim [Int. J. Numer. Meth. Engng 66 (6) (2006) 949-977], including them as type $(L,1)$. An analysis of the proposed ABCs is performed motivated by the work of Ainsworth [J. Comput. Phys. 198 (1) (2004) 106-130]. The new ABCs facilitate numerical implementations of the Helmholtz problem with ABCs if $Q_N$ finite elements are used in the physical domain. Moreover, giving more insight, the analysis presented in this work potentially aids with developing ABCs in related areas.

翻译：针对亥姆霍兹方程，我们提出了一类新型吸收边界条件。所提出的吸收边界条件可由某类简单完美匹配层推导得出，具体采用$L$层离散层，并联合使用$Q_N$拉格朗日有限元与$N$点Gauss-Legendre求积缩减积分规则。该类吸收边界条件由元组$(L,N)$分类，其反射误差阶数可达$O(R^{2LN})$（其中$R<1$）。新条件推广了Guddati与Lim [Int. J. Numer. Meth. Engng 66 (6) (2006) 949-977]提出的完美匹配离散层，后者属于$(L,1)$型情形。基于Ainsworth [J. Comput. Phys. 198 (1) (2004) 106-130]的研究工作，我们对所提出的吸收边界条件进行了理论分析。若在物理域中使用$Q_N$有限元，新条件可简化含吸收边界条件的亥姆霍兹问题的数值实现。此外，本文分析为相关领域吸收边界条件的开发提供了更深入的理论指导。