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The Poisson-Nernst-Planck (PNP) model consisting of a set of convection-diffusion equations coupled with the Poisson equation is widely used in simulations of membrane-channel protein systems. Complicated geometries (associated with poor quality mesh), complex boundary conditions and strong fixed charges commonly exist in channel protein systems, which often makes the standard finite element method fail to converge, especially for some large systems. Stabilized schemes have been studied for the convection-diffusion problem in various research fields. However, stabilization works on coupled systems like the PNP equations have been much less reported. In this work, we propose a novel stabilized finite element method for solving the three-dimensional PNP equations. A comparative study with the streamline-upwind/Petrov-Galerkin (SUPG) method is performed to examine its accuracy and numerical performance. Application to the simulation of the biological ion channel KcsA, with much more complex boundary/interface, demonstrates the capability of this method in a realistic setting. Particularly, the new method succeeds in simulating the ion channel under an unprecedented range of boundary conditions. It can deal with much higher bulk concentrations and/or membrane potentials than previous finite element methods, which greatly increases the robustness of the finite element algorithm.

computation;simulation;numerical methods;algorithm