Volume 29, Issue 2
A Diagonal Sweeping Domain Decomposition Method with Source Transfer for the Helmholtz Equation

Commun. Comput. Phys., 29 (2021), pp. 357-398.

Published online: 2020-12

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• Abstract

In this paper, we propose and test a novel diagonal sweeping domain decomposition method (DDM) with source transfer for solving the high-frequency Helmholtz equation in $\mathbb{R}^n$. In the method the computational domain is partitioned into overlapping checkerboard subdomains for source transfer with the perfectly matched layer (PML) technique, then a set of diagonal sweeps over the subdomains are specially designed to solve the system efficiently. The method improves the additive overlapping DDM [43] and the L-sweeps method [50] by employing a more efficient subdomain solving order. We show that the method achieves the exact solution of the global PML problem with $2^n$ sweeps in the constant medium case. Although the sweeping usually implies sequential subdomain solves, the number of sequential steps required for each sweep in the method is only proportional to the $n$-th root of the number of subdomains when the domain decomposition is quasi-uniform with respect to all directions, thus it is very suitable for parallel computing of the Helmholtz problem with multiple right-hand sides through the pipeline processing. Extensive numerical experiments in two and three dimensions are presented to demonstrate the effectiveness and efficiency of the proposed method.

• Keywords

Helmholtz equation, domain decomposition method, diagonal sweeping, perfectly matched layer, source transfer, parallel computing.

65N55, 65F08, 65Y05

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@Article{CiCP-29-357, author = {Leng , Wei and Ju , Lili}, title = {A Diagonal Sweeping Domain Decomposition Method with Source Transfer for the Helmholtz Equation}, journal = {Communications in Computational Physics}, year = {2020}, volume = {29}, number = {2}, pages = {357--398}, abstract = {

In this paper, we propose and test a novel diagonal sweeping domain decomposition method (DDM) with source transfer for solving the high-frequency Helmholtz equation in $\mathbb{R}^n$. In the method the computational domain is partitioned into overlapping checkerboard subdomains for source transfer with the perfectly matched layer (PML) technique, then a set of diagonal sweeps over the subdomains are specially designed to solve the system efficiently. The method improves the additive overlapping DDM [43] and the L-sweeps method [50] by employing a more efficient subdomain solving order. We show that the method achieves the exact solution of the global PML problem with $2^n$ sweeps in the constant medium case. Although the sweeping usually implies sequential subdomain solves, the number of sequential steps required for each sweep in the method is only proportional to the $n$-th root of the number of subdomains when the domain decomposition is quasi-uniform with respect to all directions, thus it is very suitable for parallel computing of the Helmholtz problem with multiple right-hand sides through the pipeline processing. Extensive numerical experiments in two and three dimensions are presented to demonstrate the effectiveness and efficiency of the proposed method.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2020-0169}, url = {http://global-sci.org/intro/article_detail/cicp/18469.html} }
TY - JOUR T1 - A Diagonal Sweeping Domain Decomposition Method with Source Transfer for the Helmholtz Equation AU - Leng , Wei AU - Ju , Lili JO - Communications in Computational Physics VL - 2 SP - 357 EP - 398 PY - 2020 DA - 2020/12 SN - 29 DO - http://doi.org/10.4208/cicp.OA-2020-0169 UR - https://global-sci.org/intro/article_detail/cicp/18469.html KW - Helmholtz equation, domain decomposition method, diagonal sweeping, perfectly matched layer, source transfer, parallel computing. AB -

In this paper, we propose and test a novel diagonal sweeping domain decomposition method (DDM) with source transfer for solving the high-frequency Helmholtz equation in $\mathbb{R}^n$. In the method the computational domain is partitioned into overlapping checkerboard subdomains for source transfer with the perfectly matched layer (PML) technique, then a set of diagonal sweeps over the subdomains are specially designed to solve the system efficiently. The method improves the additive overlapping DDM [43] and the L-sweeps method [50] by employing a more efficient subdomain solving order. We show that the method achieves the exact solution of the global PML problem with $2^n$ sweeps in the constant medium case. Although the sweeping usually implies sequential subdomain solves, the number of sequential steps required for each sweep in the method is only proportional to the $n$-th root of the number of subdomains when the domain decomposition is quasi-uniform with respect to all directions, thus it is very suitable for parallel computing of the Helmholtz problem with multiple right-hand sides through the pipeline processing. Extensive numerical experiments in two and three dimensions are presented to demonstrate the effectiveness and efficiency of the proposed method.

Wei Leng & Lili Ju. (2020). A Diagonal Sweeping Domain Decomposition Method with Source Transfer for the Helmholtz Equation. Communications in Computational Physics. 29 (2). 357-398. doi:10.4208/cicp.OA-2020-0169
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