A diagonal sweeping domain decomposition method with source transfer for the Helmholtz equation
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In this paper, we propose a novel diagonal sweeping domain decomposition method with source transfer for solving the high-frequency Helmholtz equation in R^n with n=2 or 3. In this method, the computational domain is partitioned into structured subdomains along all spatial directions (i.e., checkerboard domain decomposition) and a set of diagonal sweeps over the subdomains are specially designed to solve the global system efficiently. We prove that the proposed method achieves the exact solution 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 the method is very suitable to parallel computing for solving problems with multiple right-hand sides through the pipeline processing. Extensive numerical experiments in two and three dimensions are presented to show the effectiveness and efficiency of the proposed method.
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