A diagonal sweeping domain decomposition method with trace transfer for the Helmholtz equation
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In this paper, the diagonal sweeping domain decomposition method (DDM) for solving the high-frequency Helmholtz equation in R^n is re-proposed with the trace transfer method, where n is the dimension. The diagonal sweeping DDM uses 2^n sweeps of diagonal directions on the checkerboard domain decomposition based on the source transfer method, it is sequential in nature yet suitable for parallel computing, since the number of sequential steps is quite small compared to the number of subdomains. The advantages of changing the basic transfer method from source transfer to trace transfer are as follows: first, no overlapping region is required in the domain decomposition; second, the sweeping algorithm becomes simpler since the transferred traces have only n cardinal directions, whereas the transferred sources have all 3^n-1 directions. We proved that the exact solution is obtained with the proposed method in the constant medium case, and also in the two-layered medium case provided the source is on the same side with the first swept subdomain. The efficiency of the proposed method is demonstrated using numerical experiments in two and three dimensions, and it is found that numerical differences of the diagonal sweeping DDM with the two transfer methods are very small using second-order finite difference discretization.
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