A Wideband 2-D Fast Multipole Algorithm With a Novel Diagonalization Form

It is well known that Green’s function can be expressed by multipole expansion, plane-wave expansion, and exponential expansion (spectral representation). These three expansions constitute of the foundations of the fast multipole algorithm (FMA). The plane-wave expansion has the low-frequency breakdown issue due to its failure in capturing the evanescent spectra, while the multipole expansion is inefficient at high frequencies. The spectral representation usually involves in direction-dependent issue. In this communication, the 2-D FMA is interpreted as Parseval’s theorem in Fourier transform. To achieve a stable and accurate transition between the multipole expansion and the plane-wave expansion, a novel diagonalization in the 2-D FMA is proposed with scaled special functions based on a discrete Fourier transform. A wideband fast algorithm with high accuracies can be achieved efficiently.