Stabilization of the fast multipole method for low frequencies using multiple-precision arithmetic

We stabilize a conventional implementation of the fast multipole method (FMM) for low frequencies using multiple-precision arithmetic (MPA). We show that using MPA is a direct remedy for low-frequency breakdowns of the standard diagonalization, which is prone to numerical errors at short distances with respect to wavelength. By increasing the precision, rounding errors are suppressed until a desired level of accuracy is obtained with plane-wave expansions. As opposed to other approaches in the literature, using MPA does not require reimplementations of solvers, and it directly extends the applicability of FMM and similar methods to low-frequency problems, as well as multi-scale problems, that require globally or locally dense discretizations for accurate analysis.