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We introduce a new class of absorbing boundary conditions (ABCs) for theHelmholtz equation. The proposed ABCs can be derived from a certain simpleclass of perfectly matched layers using $L$ discrete layers and using the $Q_N$Lagrange finite element in conjunction with the $N$-point Gauss-Legendrequadrature reduced integration rule. The proposed ABCs are classified by atuple $(L,N)$, and achieve reflection error of order $O(R^{2LN})$ for some$R<1$. The new ABCs generalise the perfectly matched discrete layers proposedby 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 performedmotivated by the work of Ainsworth [J. Comput. Phys. 198 (1) (2004) 106-130].The new ABCs facilitate numerical implementations of the Helmholtz problem withABCs if $Q_N$ finite elements are used in the physical domain. Moreover, givingmore insight, the analysis presented in this work potentially aids withdeveloping ABCs in related areas.

Absorbing boundary conditions for the Helmholtz equation using Gauss-Legendre quadrature reduced integrations