Broadband Multilevel Fast Multipole Algorithm Based on an Approximate Diagonalization of the Green's Function

We present a broadband multilevel fast multipole algorithm (MLFMA) for fast and efficient solutions of three-dimensional multiscale problems involving large objects with dense discretizations. The proposed solver is based on the approximate diagonalization of the Green's function using scaled spherical and plane waves, leading to stable interaction computations for arbitrarily short distances in terms of wavelength. Despite contradictory requirements on the scaling factor that limit the accuracy of the diagonalization, the resulting low-frequency scheme is extremely stable and easy to plug into conventional MLFMA implementations by simply extending built-in tree structures, converting them into broadband solvers without needing dense programming efforts. Accuracy and efficiency of the broadband MLFMA are demonstrated on canonical problems, as well as on multiscale problems that cannot be solved efficiently via standard implementations of MLFMA.