A stabilized finite element method for the Poisson–Nernst–Planck equations in three-dimensional ion channel simulations

Abstract

The Poisson–Nernst–Planck (PNP) model characterizing the electro-diffusion process is widely used in ion channel simulations. The standard finite element method often fails to converge, due to complicated geometries, strong fixed charges and huge bias of electric potentials or ion concentrations. We propose a novel stabilized finite element method named SUPG–IP, which inherits the upwind characteristic of the streamline-upwind/Petrov–Galerkin (SUPG) method and the smoothing effect of the interior penalty method. Numerical experiments have demonstrated that the SUPG–IP method has much better performance in positivity preserving and robustness than the standard FE and SUPG methods. Especially, the benchmark simulation of the KcsA ion channel implies that the SUPG–IP method still converges successfully under an unprecedented range of membrane potentials and ion concentrations, while the SUPG method diverges.