Preconditioning based on Calderon's formulae for periodic fast multipole methods for Helmholtz' equation

Solution of periodic boundary value problems is of interest in various branches of science and engineering such as optics, electromagnetics and mechanics. In our previous studies we have developed a periodic Fast Multipole Method (FMM) as a fast solver of wave problems in periodic domains. It has been found, however, that the convergence of the iterative solvers for linear equations slows down when the solutions show anomalies related to the periodicity of the problems. In this paper, we propose preconditioning schemes based on Calderon’s formulae to accelerate convergence of iterative solvers in the periodic FMM for Helmholtz’ equations. The proposed preconditioners can be implemented more easily than conventional ones. We present several numerical examples to test the performance of the proposed preconditioners. We show that the effectiveness of these preconditioners is definite even near anomalies.