Stability of the High Frequency Fast Multipole Method for Helmholtz’ Equation in Three Dimensions

Stability limits for the high frequency plane wave expansion, which approximates the free space Green’s function in Helmholtz’ equation, are derived. This expansion is often used in the Fast Multipole Method for scattering problems in electromagnetics and acoustics. It is shown that while the original approximation of the Green’s function, based on Gegenbauer’s addition theorem, is stable except for overflows, the plane wave expansion becomes unstable due to errors from roundoff, interpolation, choice of quadrature rule and approximation of the translation operator. Numerical experiments validate the theoretical estimates.