In this paper, a complete mixed finite element method is developed for a modified Poisson–Nernst–Planck/Navier–Stokes (PNP/NS) coupling system, where the original Poisson equation in PNP system is replaced by a fourth-order elliptic equation to more precisely account for electrostatic correlations in a simplified form of the Landau–Ginzburg-type continuum model. A stabilized mixed weak form is defined for each equation of the modified PNP/NS model in terms of primary variables and their corresponding vector-valued gradient variables, based on which a stable Stokes-pair mixed finite element is thus able to be utilized to discretize all solutions to the entire modified PNP/NS model in the framework of Stokes-type mixed finite element approximation. Semi- and fully discrete mixed finite element schemes are developed and are analyzed for the presented modified PNP/NS equations, and optimal convergence rates in energy norms are obtained for both schemes. Numerical experiments are carried out to validate all attained theoretical results.

本文针对改进的Poisson-Nernst-Planck / Navier-Stokes（PNP / NS）耦合系统开发了一种完整的混合有限元方法，其中PNP系统中的原始Poisson方程由四阶椭圆方程代替。更精确地讲，以Landau-Ginzburg型连续谱模型的简化形式解释了静电相关性。根据主要变量及其对应的矢量值梯度变量，为修改后的PNP / NS模型的每个方程定义了稳定的混合弱形式，因此，可以基于该变量使用稳定的Stokes对混合有限元进行离散化在Stokes型混合有限元逼近框架内的所有改进PNP / NS模型的所有解决方案。开发了半离散和全离散混合有限元方案，并对提出的改进的PNP / NS方程进行了分析，并获得了两种方案在能量范数上的最优收敛速度。进行数值实验以验证所有获得的理论结果。