A low frequency stable plane wave addition theorem

The multilevel fast multipole algorithm (MLFMA) is a well known and very successful method for accelerating the matrix-vector products required for the iterative solution of Helmholtz problems. The MLFMA has an important drawback, namely its inability to handle scattering problems with a lot of subwavelength detail due to the low frequency (LF) breakdown of the MLFMA. There is a need to extend the MLFMA to LF, since alternative methods are less efficient (multipole methods) or hard to implement (spectral methods). In this paper a new addition theorem will be developed that does not suffer from an LF breakdown. Instead it suffers from a high-frequency (HF) breakdown. The new method relies on a novel set of distributions, the so-called pseudospherical harmonics, closely related to the spherical harmonics. These allow the discretization points and translation operators to be calculated in closed form. Hence the method presented in this paper allows the easy implementation of a method that is stable at LF. Furthermore, a combination of the traditional MLFMA and the new method allows for the construction of a broadband MLFMA.