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.

在本文中，我们为层状电解质-介电介质中的3-D线性化泊松-玻尔兹曼（PB）方程提出了一种快速多极方法（FMM）。我们将把先前在层状介质[1]，[2]中用于Helmholtz和Laplace方程的FMM的工作扩展到电解质和介电层状介质的情况，以用于由生物物理和胶体流体产生的应用，例如离子通道传输和Helmholtz双层。为此，开发了两个关键的数学公式：首先，推导了纯虚波数的Funk-Hecke公式，该公式有助于推导层状电解质-介电介质中电荷的潜在远场的多极膨胀。其次，构造了一个递归公式，用于FMM算法中使用的Sommerfeld型积分的运行时计算。$Ø（ñ）$ 计算复杂性是自由空间中电荷相互作用的经典FMM。