Moving Mesh Methods For A Class Of Poisson-Nernst-Planck Equations

Poisson-Nernst-Planck(PNP)equations are coupled partial differential equations,which mainly describe the electrostatic diffusion reaction of biomolecules,and are theoretical models to simulate the permeation mechanism of ion channels.When the PNP equation is applied to the ion channel problem,it will produce singularity because there are many charges on the membrane interface,and its numerical solution usually requires a higher degree of freedom.The moving grid method can achieve the required precision with fewer grid points,keep the total number of grid points unchanged,and make the grid points concentrate in the region of rapid change of solution or derivative through node movement.The mobile grid method used in this paper can be divided into computing domain based mobile grid method and physical domain based mobile grid method according to the variational principle.This paper mainly studies the calculation of moving grid method for a class of PNP equations,including three aspects.Firstly,for a class of typical time-dependent PNP equations,the moving grid finite element method based on computing domain is designed.The numerical results of one-dimensional time-dependent PNP equations with singularity show that the error of L2 mode of the moving grid finite element method based on computing domain is significantly smaller than that of the uniform grid finite element method.Secondly,for a class of typical time-dependent PNP equations,a moving grid finite element method based on physical domain is designed to solve three examples of time-dependent PNP equations,namely,the smooth solution of two-dimensional time-dependent PNP equations,one and two dimensional time-dependent PNP equations with singularity.The numerical results of two-dimensional time-dependent PNP equations for smooth solutions show that the modulus error of L2 reaches the optimal order and the grid movement is smooth.The numerical results of one-dimensional time-dependent PNP equations with singularity show that the moving grid finite element method based on physical domain has smaller L2 mode error and captures singularity compared with the uniform grid finite element method.The two dimensional time-dependent PNP equations with singularity show that the moving grid finite element method based on physical domain can better fit the exact solution image and capture the singularity,while the numerical solution of the uniform grid finite element method is almost opposite to the exact solution image.Finally,in order to improve the stability of numerical calculation of PNP equations,an edge-average finite element discrete scheme is constructed for a class of nonlinear PNP equations.The stiffness matrix of this scheme under appropriate grid requirements is an M-matrix,and the numerical solution stability is good.The numerical results of the three-dimensional nonlinear PNP equation show that compared with the standard finite element method,the edge-average finite element method takes about one third of the CPU time,and the L2 mode error is slightly smaller and reaches the optimal order.In addition,for two types of PNP equations,the moving grid edge mean finite element method based on physical domain is designed.The numerical results of one-dimensional time-dependent typical PNP equations and nonlinear PNP equations with singularity show that the moving grid edge mean finite element method has less CPU time and smaller L2 mode error than the uniform grid finite element method.