Restrictions on the use of sweeping type preconditioners for Helmholtz problems

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.