Broadband implementation of MLFMA using approximate scaled diagonalization of the Green's function

It is well known that the conventional diagonalization used in the multilevel fast multipole algorithm (MLFMA) suffers from low-frequency breakdowns, which limit the application of this powerful method to multiscale problems involving large objects with small (but important) details in terms of wavelength. Using multipoles explicitly or resorting to alternative expansions incorporating evanescent waves for subwavelength interactions are common techniques for broadband implementations of MLFMA, while all these techniques need extensive efforts and re-programming. Recently, we presented an approximate diagonalization of the three-dimensional Green's function using scaled Hankel functions and plane waves (Ö. Ergül and B. Karaosmanoğlu, IEEE Antennas Wireless Propag. Lett., 13, 2014, pp. 1054–1056). While its accuracy is limited (e.g., 3–4% in the worst case for the challenging one-box-buffer scheme), the approximate diagonalization provides stable computations of interactions at arbitrarily short distances. As a major advantage, the approximate diagonalization needs only minor modifications in the existing codes, including the parallel programs, to stabilize the conventional implementations for low frequencies.