Absorbing boundary conditions for the Helmholtz equation using Gauss-Legendre quadrature reduced integrations
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866914784916537344 |
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| author | Sagiyama, Koki |
| author_facet | Sagiyama, Koki |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2308_12255 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Absorbing boundary conditions for the Helmholtz equation using Gauss-Legendre quadrature reduced integrations Sagiyama, Koki Numerical Analysis Mathematical Physics 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. |
| title | Absorbing boundary conditions for the Helmholtz equation using Gauss-Legendre quadrature reduced integrations |
| topic | Numerical Analysis Mathematical Physics |
| url | https://arxiv.org/abs/2308.12255 |