Multiple-precision MLFMA for efficient and accurate solutions of broadband electromagnetic problems

The multilevel fast multipole algorithm (MLFMA) is a popular full-wave electromagnetic solver that enables the solution of electrically large problems with an extremely large number of unknowns. As with all computational electromagnetics solvers, active research is ongoing to extend the limitations of MLFMA for larger problems with finer geometrical details. For electrically small structures MLFMA suffers from the low-frequency breakdown, while more efficient schemes are required for electrically larger problems. We propose and demonstrate an elegant solution to the aforementioned problems by introducing a multiple-precision arithmetic (MPA) framework to the inherent hierarchical tree structure of MLFMA, dubbed the multiple-precision multilevel fast multipole algorithm (MP-MLFMA). With the introduction of MPMLFMA we show that a distinct machine precision can be assigned to each level of the tree structure of MLFMA, which enables accurate and efficient solutions of problems with deep tree-structures over arbitrarily large frequency bandwidths. To determine the required machine precisions for a given tree structure, as well as the number of harmonics required for an accurate error control of the translation operator of MP-MLFMA, we introduce and validate a novel error control scheme with accurate design curves that is valid at all frequencies, for the first time in the literature. Combined with the proposed error control scheme, we present the capabilities of MP-MLFMA over a wide range of broadband and deep tree-structure scattering problems. We also illustrate the true potential efficiency of MP-MLFMA, with a simple MPA framework implementation on a single-precision processor. With the hardware-defined implementation, we show the super-linear speed-up potential of MP-MLFMA for low-precisions.