Accelerating fast multipole methods for the Helmholtz equation at low frequencies

The authors describe a diagonal form for translating far-field expansions to use in low frequency fast multipole methods. Their approach combines evanescent and propagating plane waves to reduce the computational cost of FMM implementation. More specifically, we present the analytic foundations for a new version of the fast multipole method for the scalar Helmholtz equation in the low frequency regime. The computational cost of existing FMM implementations, is dominated by the expense of translating far field partial wave expansions to local ones, requiring 189p⁴ or 189p³ operations per box, where harmonics up to order p² have been retained. By developing a new expansion in plane waves, we can diagonalize these translation operators. The new low frequency FMM (LF-FMM) requires 40p²+6p² operations per box