An Effective Finite Element Iterative Solver for a Poisson--Nernst--Planck Ion Channel Model with Periodic Boundary Conditions
Dexuan Xie, Benzhuo Lu
Abstract
A system of Poisson--Nernst--Planck equations (PNP) is an important dielectric continuum model for simulating ion transport across biological membrane. In this paper, a PNP ion channel model with periodic boundary value conditions, denoted by PNPic, is presented and solved numerically with an effective finite element iterative method. In particular, the periodic boundary value conditions are used to mimic an infinitely large ion channel membrane, and the PNPic finite element solver includes (1) a PNPic solution decomposition scheme for overcoming the singularity difficulty caused by atomic charges, (2) Slotboom variables for transforming each related Nernst--Planck equation to avoid gradient calculation for any electrostatic potential function, (3) an efficient modified Newton iterative algorithm for solving each related nonlinear finite element equation, and (4) communication operators for carrying out functions operations between different finite element function spaces. This effective PNPic solver is implemented as a software package based on the state-of-the-art finite element library from the FEniCS project and an ion channel mesh generation package developed in Lu's group. Numerical results demonstrate the convergence of the PNPic finite element iterative solver and the performance of the PNPic software package. Moreover, the PNPic model is validated by the cation selectivity property and electric current experimental data of an ion channel protein.