A Stabilized Finite Element Method for Modified Poisson-Nernst-Planck Equations to Determine Ion Flow Through a Nanopore (Communications in Computational Physics, 2014)
The conventional Poisson-Nernst-Planck equations do not account for thefinite size of ions explicitly. This leads to solutions featuring unrealistically high ionicconcentrations in the regions subject to external potentials, in particular, near highlycharged surfaces. A modified form of the Poisson-Nernst-Planck equations accountsfor steric effects and results in solutions with finite ion concentrations. Here, we evaluate numerical methods for solving the modified Poisson-Nernst-Planck equations bymodeling electric field-driven transport of ions through a nanopore. We describe anovel, robust finite element solver that combines the applications of the Newton'smethod to the nonlinear Galerkin form of the equations, augmented with stabilizationterms to appropriately handle the drift-diffusion processes.To make direct comparison with particle-based simulations possible, our method isspecifically designed to produce solutions under periodic boundary conditions andto conserve the number of ions in the solution domain. We test our finite elementsolver on a set of challenging numerical experiments that include calculations of theion distribution in a volume confined between two charged plates, calculations of theionic current though a nanopore subject to an external electric field, and modeling theeffect of a DNA molecule on the ion concentration and nanopore current.