Optimized Schwarz Algorithm with Double Sweep Preconditioner for the Helmholtz Equation

In previous work (Y. Boubendir, X. Antoine and C. Geuzaine, A quasi-optimal nonoverlapping domain decomposition algorithm for the Helmholtz equation, JCP, 231, pp. 262-280, 2012), it was shown how a Domain Decomposition Method (DDM) formulation of the Helmholtz problem, using impedance-matching boundary conditions, can be set up and accelerated with a Krylov solver. This optimized Schwarz algorithm involves the so-called Dirichlet-to-Neumann (DtN) map for the transmission of information between subdomains, which can be approximated in different ways. While the quality of this approximation is critical to the rate of convergence of the algorithm, recent works have succesfully addressed this problem. Another difficulty with this algorithm resides in the spectral properties of the iteration operator, that still prevent fast convergence in the case of many subdomains and a layered decomposition.