Parallel Double Sweep Preconditioner for the Optimized Schwarz Algorithm Applied to High Frequency Helmholtz and Maxwell Equations

Abstract

The principle of sweeping to accelerate the solution of wave propagation problems has recently retained much interest, yet with different approaches (Engquist and Ying, Multiscale Model Simul 9(2):686–710, 2011; Stolk, J Comput Phys 241:240–252, 2013). We recently proposed a preconditioner for the optimized Schwarz algorithm, based on a propagation of information using a double sequence of subproblems solves, or sweeps (Vion et al., A DDM double sweep preconditioner for the Helmholtz equation with matrix probing of the DtN map, Mathematical and Numerical Aspects of Wave Propagation WAVES 2013, June 2013; Vion and Geuzaine, J Comput Phys, 2014, Preprint, submitted). Although this procedure significantly reduces the number of iterations when many subproblems are involved, the sequential nature of the process hinders the scalability of the algorithm on parallel computer architectures. Here we propose a modified version of the algorithm that concurrently runs partial sweeps on smaller groups of domains, which efficiently reduces the preconditioner application time on parallel machines. We show that the algorithm is applicable to both Helmholtz and Maxwell equations.