Resonance frequencies are complex eigenvalues at which the homogeneous transmission problems have non-trivial solutions. These frequencies are of interest because they affect the behavior of the solutions even when the frequency is real. The resonance frequencies are related to problems for infinite domains which can be solved efficiently with the Boundary Integral Equation Method (BIEM). We thus consider a numerical method of determining resonance frequencies with fast BIEM and the SakuraiSugiura projection Method (SSM). However, BIEM may have fictitious eigenvalues even when one uses Müller or PMCHWT formulations which are known to be resonance free when the frequency is real valued.In this paper, we propose new BIEs for transmission problems with which one can distinguish true and fictitious eigenvalues easily. Specifically, we consider waveguide problems for the Helmholtz equation in 2D and standard scattering problems for Maxwell's equations in 3D. We verify numerically that the proposed BIEs can separate the fictitious eigenvalues from the true ones in these problems. We show that the obtained true complex eigenvalues are related to the behavior of the solution significantly. We also show that the fictitious eigenvalues may affect the accuracy of BIE solutions in standard boundary value problems even when the frequency is real.

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.

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SUMMARYPreconditioning methods based on Calderon's formulae for the periodic fast multipole method for elastodynamics in 3D are investigated. Three different types of formulations are proposed. The first type is a preconditioning just by appropriately ordering the coefficient matrix without multiplying preconditioners. The other two types utilise preconditioners constructed using matrices needed in the main fast multipole method algorithms. We make several numerical experiments with proposed preconditioners to confirm the efficiency of these proposed methods. We also conclude that the preconditioning of the first type is faster with respect to the computational time than other preconditioning methods discussed in this article. Copyright