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

TL;DR

This paper introduces a new class of absorbing boundary conditions for the Helmholtz equation using Gauss-Legendre quadrature reduced integrations, improving reflection error rates and generalizing previous methods.

Contribution

It proposes a novel class of ABCs based on layered discretization and quadrature rules, extending existing methods and providing analytical insights for numerical Helmholtz problem solutions.

Findings

Achieves reflection error of order O(R^{2LN}) with R<1

Generalizes previous perfectly matched discrete layers

Facilitates numerical implementation with finite elements

Abstract

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.