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A Low-Frequency Stable Broadband Multilevel Fast Multipole Algorithm Using Plane Wave Multipole Hybridization

We propose a broadband multilevel fast multipole algorithm using hybridization of the plane wave and the multipole expansions of Green’s function in the analysis of 3-D electromagnetic problems. In the proposed method, the diagonal plane-wave expansion is used for low order harmonics, which captures most of the propagating spectrums, while the dense multipole expansion is used for high order harmonics, which captures most of the evanescent spectrums. By analyzing the errors and choosing the numbers of harmonics accordingly in the plane wave and the multipole expansions, the method is free of low-frequency breakdown and remains accurate at arbitrarily low frequencies, which is similar to the dense fast multipole algorithm (FMA). Meanwhile, the method reduces to the conventional diagonal FMA as the frequency increases. Therefore, it can be regarded as the generalization of the conventional dense and diagonal FMAs. Numerical studies in this paper show that very high accuracies can be achieved using the method.