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

TL;DR

This work introduces a new class of absorbing boundary conditions for the Helmholtz equation that leverages $L$ discrete artificial layers, $Q_N$ finite elements, and $N$-point Gauss-Legendre reduced integration, classified as $(L,N)$. The proposed ABCs generalize PMDLs (type $(L,1)$) and achieve a reflection error that decays like $O(R^{2LN})$ for some $R<1$, while preserving a monolithic discretization across the physical and artificial domains. The analysis, inspired by Ainsworth, reveals a Padé-approximation structure via ${}_1F_1^N$ functions, linking the reflection to the zeros of $[N/N]_{\, ext{exp}(-z)}$ and enabling near-exact Sommerfeld behavior with transformed boundary conditions. Numerical experiments in 1D and 3D confirm the theoretical predictions, showing rapid convergence and accurate absorption for a range of $(L,N)$ choices, and highlighting the practical viability of high-order ABCs in complex geometries.

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.

Figures (5)

• Figure 1: Reflection coefficient computed for the one-dimensional Helmholtz problem with $\gamma\in\{z\in\mathbb{C}|0<\text{Re}(z)<8,-8<\text{Im}(z)<+8\}$ in $(-1,0)$ with the ABC of type $(1,N)$ with $\gamma_1=1$ applied in $(0,1)$, where $N\in\{1,2,3,4\}$. A $Q_N$ Lagrange finite element with $N$-point reduced integration was used in $(-1,0)$ as well as in $(0,1)$. For each $N\in\{1,2,3,4\}$, the reflection coefficient virtually vanished where $\gamma/\gamma_1$ coincided with the zeros of $[N/N]_{\exp{(-z)}}$; those points are shown with the $+$ symbols.
• Figure 2: Reflection coefficient computed for the one-dimensional Helmholtz problem with $\gamma\in\{z\in\mathbb{C}|0<\text{Re}(z)<8,-8<\text{Im}(z)<+8\}$ in $(-1,0)$ with the ABC of type $(1,N)$ with $\gamma_1=1/2+\sqrt{3}/2\!\cdot\!\mathfrak{i}$ applied in $(0,1)$, where $N\in\{1,2,3,4\}$. A $Q_N$ Lagrange finite element with $N$-point reduced integration was used in $(-1,0)$ as well as in $(0,1)$. For each $N\in\{1,2,3,4\}$, the reflection coefficient virtually vanished where $\gamma/\gamma_1$ coincided with the zeros of $[N/N]_{\exp{(-z)}}$; those points are shown with the $+$ symbols.
• Figure 3: Exact solution to the example Helmholtz problem with $s=+4.00+0.25\mathfrak{i}$.
• Figure 5: Exact solution to the example Helmholtz problem with $s=+0.25+4.00\mathfrak{i}$.
• Figure 7: Exact solution to the example Helmholtz problem with $s=+4.00\mathfrak{i}$.