A multipole based treecode using spherical harmonics for potentials of the form r -λ

TL;DR: The relationship between Gegenbauer and Legendre polynomials is exploited to come up with a natural generalization of the multipole expansion theorem used in the classical fast multipole algorithm that is used with a hierarchical scheme to compute the potentials.

Abstract: In this paper we describe an efficient algorithm for computing the potentials of the form r$^{\rm -{\it \lambda}}$ where λ ≥ 1. This treecode algorithm uses spherical harmonics to compute multipole coefficients that are used to evaluate these potentials. The key idea in this algorithm is the use of Gegenbauer polynomials to represent r$^{\rm -{\it \lambda}}$ in a manner analogous to the use of Legendre polynomials for the expansion of the Coulomb potential r−1. We exploit the relationship between Gegenbauer and Legendre polynomials to come up with a natural generalization of the multipole expansion theorem used in the classical fast multipole algorithm [2]. This theorem is used with a hierarchical scheme to compute the potentials. The resulting algorithm has known error bounds and can be easily implemented with modification to the existing fast multipole algorithm. The complexity of the algorithm is O(p3N log N) and has several advantages over the existing Cartesian coordinates based expansion schemes.