A Two-Grid Method For The Nonlinear Steady-State Poisson-Nernst-Planck Equations

This paper mainly studies the two-grid finite element algorithm for the nonlinear steady-state Poisson-Nernst-Planck(PNP)equation.The PNP equation is a type of the partial differential equation coupling system,which is used to describe the solute biomolecular systems electrodiffusion of the mobile ions in the medium.The research content of this article has the following three parts.Firstly,for a class of nonlinear steady-state PNP equations,theH~1 norm error estimate of the finite element is given,and the correctness of the theory is verified by the numerical experiments.Secondly,for this kind of nonlinear steady-state PNP equations,two types of two-grid finite element algorithms,semi-decoupled and fully decoupled,are designed,and the corresponding convergence analysis is given.Compared with the standard finite element method,the two-grid finite element algorithm can achieve the decoupling of the equations and speed up the solution of the nonlinear steady-state PNP equation,but the convergence order is still the same as the finite element method.Numerical experiments verify the theoretical results,and show that the two-grid finite element method can greatly save the computational time and improve the computational efficiency.Thirdly,for a class of steady-state PNP equations,using the~2L projection operators,the superconvergence error analysis of the PNP equation solutions is given.Numerical experiments show that the precision of finite element solution can be improved by post-processing the finite element solution with the ~2L projection.