A Diagonal Sweeping Domain Decomposition Method with Source Transfer for the Helmholtz Equation (Communications in Computational Physics, 2020)
In this paper, we propose and test a novel diagonal sweeping domaindecomposition method (DDM) with source transfer for solving the high-frequencyHelmholtz equation in $\mathbb{R}^n$. In the method the computational domain is partitioned intooverlapping checkerboard subdomains for source transfer with the perfectly matchedlayer (PML) technique, then a set of diagonal sweeps over the subdomains are speciallydesigned 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 globalPML problem with $2^n$ sweeps in the constant medium case. Although the sweepingusually implies sequential subdomain solves, the number of sequential steps requiredfor each sweep in the method is only proportional to the $n$-th root of the number ofsubdomains when the domain decomposition is quasi-uniform with respect to all directions, thus it is very suitable for parallel computing of the Helmholtz problem withmultiple right-hand sides through the pipeline processing. Extensive numerical experiments in two and three dimensions are presented to demonstrate the effectivenessand efficiency of the proposed method.