We introduce a new class of absorbing boundary conditions (ABCs) for the Helmholtz equation. The proposed ABCs are obtained by using $L$ discrete layers and the $Q_N$ Lagrange finite element in conjunction with the $N$-point Gauss-Legendre quadrature reduced integration rule in a specific formulation of perfectly matched layers. 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 as well as give more insight into this field for the further advancement.

翻译：本文针对Helmholtz方程提出了一类新的吸收边界条件。所提出的吸收边界条件通过采用$L$个离散层、$Q_N$拉格朗日有限元，并结合完美匹配层特定公式中的$N$点Gauss-Legendre求积降阶积分规则获得。所提出的吸收边界条件由元组$(L,N)$分类，并在某$R<1$条件下实现$O(R^{2LN})$阶反射误差。新吸收边界条件推广了Guddati与Lim [Int. J. Numer. Meth. Engng 66 (6) (2006) 949-977]提出的完美匹配离散层，将其作为类型$(L,1)$纳入其中。受Ainsworth [J. Comput. Phys. 198 (1) (2004) 106-130]工作的启发，对所提出的吸收边界条件进行了分析。当物理域中使用$Q_N$有限元时，新吸收边界条件有助于带吸收边界条件的Helmholtz问题的数值实现，并为该领域的进一步发展提供更深入的见解。