Abstract
Sweeping type preconditioners have become a focus of attention for solving high frequency time harmonic wave propagation problems. These methods can be found under various names in the literature: in addition to sweeping, one finds the older approach of the Analytic Incomplete LU (AILU), optimized Schwarz methods, and more recently also source transfer domain decomposition, method based on single layer potentials, and method of polarized traces. An important innovation in sweeping methods is to use perfectly matched layer (PML) transmission conditions. In the constant wavenumber case, one can approximate the optimal transmission conditions represented by the Dirichlet to Neumann operator (DtN) arbitrarily well using large enough PMLs. We give in this short manuscript a simple, compact representation of these methods which allows us to explain exactly how they work, and test what happens in the case of non-constant wave number, in particular layered media in the difficult case where the layers are aligned against the sweeping direction. We find that iteration numbers of all these methods remain robust for very small contrast variations, in the order of a few percent, but then deteriorate, with linear growth both in the wave number as well as in the number of subdomains, as soon as the contrast variations reach order one.
The author “H. Zhang” was supported by ZJOU Research Start Fund.
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Notes
- 1.
Provided the domain has indeed an open end or such a high order PML on the side where the sweeping begins.
- 2.
It is the exact Schur complement, including all boundary information, the only approximation is the constant wave number.
- 3.
The boundary points are not plotted, so one cannot see the homogeneous Dirichlet condition.
- 4.
There are also two interesting apparent anomalies: in the smaller wavenumber case, for p = 4 and α = 0.05 (and also one in the larger wave number case), the spectral radius is bigger than one, but for the source term f = 1 we observe convergence. We iterated in this case however further, and then the iterations also start to diverge, it is only that the divergent modes are not stimulated at the beginning by the source term f = 1 and zero initial guess, a typical phenomenon known from power iterations, which explains in the table the general observation that the problem with f = 1 is easier to solve than with the other sources, also for GMRES. For the same p = 4 and α = 0.1, we then get surprisingly a spectral radius again smaller than 1, which is a lucky configuration and not observed for more subdomains or different α.
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Gander, M.J., Zhang, H. (2018). Restrictions on the Use of Sweeping Type Preconditioners for Helmholtz Problems. In: Bjørstad, P., et al. Domain Decomposition Methods in Science and Engineering XXIV . DD 2017. Lecture Notes in Computational Science and Engineering, vol 125. Springer, Cham. https://doi.org/10.1007/978-3-319-93873-8_30
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