Fast multipole method for 3-D Poisson-Boltzmann equation in layered electrolyte-dielectric media

Abstract

In this paper, we propose a fast multipole method (FMM) for 3-D linearized Poisson–Boltzmann (PB) equation in layered electrolyte-dielectric media. We will extend our previous work on FMMs for Helmholtz and Laplace equations in layered media [1,2] to the case of electrolyte and dielectric layered media for applications arising from biophysics and colloidal fluids, such as ion channel transport and Helmholtz double layers. Two key mathematical formulas are developed for this purpose: Firstly, a Funk–Hecke formula for purely imaginary wave numbers is derived, which facilitates the derivation of multipole expansions of the potential far fields of charges in layered electrolyte-dielectric media. Secondly, a recurrence formula is constructed for run-time computations of the Sommerfeld-type integrals used in the FMM algorithm. Numerical results show that the proposed FMM for interactions of charges embedded in layered media under screened PB potentials has the same accuracy and the O(N) computational complexity as the classic FMM for charge interactions in the free space.