Fast Multipole Method Using the Cauchy Integral Formula

The fast multipole method (FMM) is a technique allowing the fast calculation of long-range interactions between

N

points in

O

(

N

) or

O

(

N

ln

N

) steps with some prescribed error tolerance. The FMM has found many applications in the field of integral equations and boundary element methods, in particular by accelerating the solution of dense linear systems arising from such formulations. Standard FMMs are derived from analytic expansions of the kernel, for example using spherical harmonics or Taylor expansions. In recent years, the range of applicability and the ease of use of FMMs has been extended by the introduction of black box (Fong and Darve, Journal of Computational Physics 228:8712–8725, 2009) or kernel independent techniques (Ying, Biros and Zorin, Journal of Computational Physics 196:591–626, 2004). In these approaches, the user only provides a subroutine to numerically calculate the interaction kernel. This allows changing the definition of the kernel with minimal change to the computer program. This paper presents a novel kernel independent FMM, which leads to diagonal multipole-to-local operators. This results in a significant reduction in the computational cost (Fong and Darve, Journal of Computational Physics 228:8712–8725, 2009), in particular when high accuracy is needed. The approach is based on Cauchy’s integral formula and the Laplace transform. We will present a short numerical analysis of the convergence and some preliminary numerical results in the case of a single level one dimensional FMM.