Two-grid And Adaptive Finite Element Methods For Poisson-nernst-planck Equations

Poisson-Nernst-Planck(PNP)equations are strongly coupled nonlinear par-tial differential equations,which are composed of the Poisson equation and Nernst-Planck equations.This model has been widely used to describe the electrostatic diffusion-reaction process in biochemistry,the transport of charged particles in semiconductors,ion conversion in biological cell membranes and other applica-tions areas.Finite element method is a popular discretization method for solving the PNP equations.Therefore,it is of great theoretical and practical significance to study the finite element error estimates and its fast numerical algorithms for the PNP equations.There are mainly three contributes in this paper shown as follows.For the classic time-dependent PNP equations which describe the mass con-servation of ions,by introducing a suitable finite element projection operator,the optimal~2error estimate of the semi-discrete linear finite element scheme is pre-sented first.Then the optimal~2error estimate is also established for a backward Euler fully discrete linear finite element scheme,which is widely used and can better preserve the physical properties of the PNP equations.The correctness of the theoretical results is verified by some numerical experiments.The semi-and full decoupled schemes of the two-grid method are established for the fully discrete finite element method of the time-dependent PNP equation.Based on the decoupling ideas,the computational system is decoupled to smaller systems by using coarse space solution as a reliable approximation solution on the fine space at each time level,which can speed up the solution process compared with the standard finite element method.Based on the optimal~2error estimate of the finite element solution derived in this paper,the optimal~1error estimate of the two-grid finite element solution is obtained for the electrostatic potential.The~2and~1error estimates of the two grid finite element solutions for the concentrations are given.The numerical experiments show that if the mesh sizeand?satisfy=(?~12),the two-grid method can retain the same convergence order as the standard finite element method.A series of numerical experiments including a practical ion channel problem verify the effectiveness of the two-grid algorithms,and it is shown by less CPU time that the efficiency of finite element method has been much improved by using the two-grid algorithms.For the linear finite element scheme of a class of modified nonlinear steady-state PNP model,the gradient recovery type a posteriori error estimates are dis-cussed.For the first time,a strict posterior error analysis is carried out for the model.The upper bounds and lower bounds of the a posteriori error estimators are obtained for both the electrostatic potential and concentrations,and an adaptive finite element algorithm based on the a posterior error estimators is designed.In particular,the a posteriori error estimators derived in this paper are still applicable to the classic steady-state PNP equations.Several numerical experiments verify the reliability and effectiveness.The adaptive computing improves the efficiency of solving the singular PNP problem.