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