Fast Evaluation of Derivatives of Green's Functions Using Recurrences

High-order derivatives of Green's functions are a key ingredient in Taylor-based fast multipole methods, Barnes-Hut $n$-body algorithms, and quadrature by expansion (QBX). In these settings, derivatives underpin either the formation, evaluation, and/or translation of Taylor expansions. In this article, we provide hybrid symbolic-numerical procedures that generate recurrences to attain an $O(n)$ cost for the the computation of $n$ derivatives (i.e. $O(1)$ per derivative) for arbitrary radially symmetric Green's functions. These procedures are general--only requiring knowledge of the PDE that the Green's function solves. We show that the algorithm has controlled, theoretically-understood error. We apply these methods to the method of quadrature by expansion, a method for the evaluation of singular layer potentials, which requires higher-order derivatives of Green's functions. In doing so, we contribute a new rotation-based method for target-specific QBX evaluation in the Cartesian setting that attains dramatically lower cost than existing symbolic approaches. Numerical experiments support our claims of accuracy and cost.