Maxwell Cartesian Harmonics and the Static Fast Multipole Method

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The static fast multipole method (FMM) has revolutionized the computation of pairwise interaction in N-body systems. This technique has been applied to diverse fields ranging from astrophysics to biochemistry to electrical engineering. The development of static FMM has progressed along two fronts; (i) expansions that are based on tesseral harmonics and (ii) Taylor series expansions. Indeed, the history of an FMM scheme based on the latter can be traced back to the same time period as the former. It was proven that while for a one-level implementation the computational complexity of the former scheme scales as O(p2N) the latter scales as O(p3N), where p is the number of harmonics used and N is the number of source/observer points. The interest in revisiting this scheme comes from the following observations: (i) Taylor's expansion involves representing the fields in terms of Cartesian tensors; (ii) there is an intimate connection between Cartesian tensors and the spherical harmonics. Indeed, it is well known that the following statements hold true: (i) the components of a traceless tensor of rank n serve as constant coefficients in a spherical harmonic of degree n, and (ii) there is a class of traceless tensors of rank n whose components are n-degree spherical harmonics functions of x, y, z. These connections imply that there should be an intimate connection between the two seemingly disparate schemes, and one should be able to obtain a similar cost structure for both methods. This connection is explored further in this paper. The next section briefly introduces the notation that is used in the rest of this paper. Next is the derivation of the steps necessary to implement the multilevel level FMM. Due to the paucity of space, several theorems (both this paper's and others) are stated without proofs