Block-Jacobi sweeping preconditioners for optimized Schwarz methods applied to the Helmholtz equation
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The parallel performances of sweeping-type algorithms for high-frequency time-harmonic wave problems have been recently improved by departing from standard layer-type domain decomposition and introducing a new sweeping strategy on a checkerboard-type domain decomposition, where sweeps can be performed more flexibly. These sweeps can be done by a certain number of steps, each of which provides the necessary information from subdomains on which solutions have been obtained to their next neighboring subdomains. Although, subproblems in these subdomains can be solved concurrently at each step, the sequential nature of the process of the sweeping approaches still exists, which limits their parallel performances. Moreover, the sweeping approaches can be interpreted as a completely approximate LU factorization, which implies a huge computation cost. We propose block-Jacobi sweeping preconditioners, which are improved variants of sweeping-type preconditioners. The new feature of these improved variants can be interpreted as several partial sweeps, which can be performed parallelly. We present several two- and three-dimensional finite element results with constant and various wave speeds to study and compare the original and block-Jacobi sweeping preconditioners.
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