| The Poisson-Nernst-Planck(PNP)model is mainly used to describe the electrostatic diffusion reaction process in molecular solutions.It is widely used in physics,biology,chemistry,engineering and other fields.There are singularities in the PNP equations in practical problems.In order to solve the singularity problem,we use the adaptive finite element method to solve the time-dependent PNP equations,and the a posteriori error analysis is a key part of the adaptive finite element method.Therefore,we do the following research on the time-dependent PNP equations:First,for the time-dependent PNP equations,the backward Euler scheme is applied to time discretization,and the linear triangular finite element is applied to space discretization.For this discrete scheme,we construct residual type a posteriori error indicators,which consist of space residual indicators and time residual indicators.The posteriori error analysis of the two types of solutions of the potential and ion concentration in the PNP equations is performed,and the results of the upper and lower bounds of the error are given.Secondly,based on the residual posteriori error indicators constructed in the error analysis,an adaptive finite element algorithm for solving the time-dependent PNP equations is designed,and three numerical examples are given in the Fortran environment.The numerical results show that the adaptive finite element algorithm based on the residual type a posteriori error indicators are effective and reliable in solving the singularity problem of the time-dependent PNP equations.Finally,we use the Crank-Nicolson scheme to discretize the time-dependent PNP equations,and introduce a new Ritz projection operator.Using this projection operator,the priori error analysis of the finite element discretization of the time-dependent PNP equations.We obtain the optimal error estimates for the finite element solutions in theL~2norm.Finally,a numerical experiment is given to verify the theoretical result. |
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