Poisson–Nernst–Planck equations for simulating biomolecular diffusion–reaction processes I: Finite element solutions

In this paper we developed accurate finite element methods for solving 3-D Poisson–Nernst–Planck (PNP) equations with singular permanent charges for simulating electrodiffusion in solvated biomolecular systems. The electrostatic Poisson equation was defined in the biomolecules and in the solvent, while the Nernst–Planck equation was defined only in the solvent. We applied a stable regularization scheme to remove the singular component of the electrostatic potential induced by the permanent charges inside biomolecules, and formulated regular, well-posed PNP equations. An inexact-Newton method was used to solve the coupled nonlinear elliptic equations for the steady problems; while an Adams–Bashforth–Crank–Nicolson method was devised for time integration for the unsteady electrodiffusion. We numerically investigated the conditioning of the stiffness matrices for the finite element approximations of the two formulations of the Nernst–Planck equation, and theoretically proved that the transformed formulation is always associated with an ill-conditioned stiffness matrix. We also studied the electroneutrality of the solution and its relation with the boundary conditions on the molecular surface, and concluded that a large net charge concentration is always present near the molecular surface due to the presence of multiple species of charged particles in the solution. The numerical methods are shown to be accurate and stable by various test problems, and are applicable to real large-scale biophysical electrodiffusion problems.