论文标题
对泊松 - nernst-planck/navier- stokes方程的耦合和无差的虚拟元素方法的最佳错误估计
Optimal error estimates of coupled and divergence-free virtual element methods for the Poisson--Nernst--Planck/Navier--Stokes equations
论文作者
论文摘要
在本文中，我们提出和分析了用于求解耦合的Poisson poisson-nernst-Planck（PNP）和Navier-Stokes-Stokes（NS）方程的完全耦合，非线性和能量稳定的虚拟元素方法（VEM）（VEM）建模微型流体和电化学系统（在电源系统中的差异）（通过Incompress Incompress compompress coutpls couours couours couou）。使用混合VEM来离散NS方程，而原始形式的经典VEM用于离散PNP方程。固定点理论证明了相关VEM解决方案的稳定性，存在和独特性。该计划的全球质量保护和电能衰减也得到了证明。此外，我们获得了$ h^{1} $ - 标准中PNP方程的静电电位和离子浓度的无条件最佳误差估计，以及$ \ \ m athbf {h}^{1} $ - 和$ l^{2} $的NS方程的速度和压力。最后，提出了几个数值实验，以支持收敛的理论分析，并说明该方法在模拟离子流体中电动不稳定性的开始时的令人满意的性能，并研究它们如何受离子浓度和施加电压的不同值的影响。这些测试涉及水的淡化中的应用。
In this article, we propose and analyze a fully coupled, nonlinear, and energy-stable virtual element method (VEM) for solving the coupled Poisson-Nernst-Planck (PNP) and Navier--Stokes (NS) equations modeling microfluidic and electrochemical systems (diffuse transport of charged species within incompressible fluids coupled through electrostatic forces). A mixed VEM is employed to discretize the NS equations whereas classical VEM in primal form is used to discretize the PNP equations. The stability, existence and uniqueness of solution of the associated VEM are proved by fixed point theory. Global mass conservation and electric energy decay of the scheme are also proved. Also, we obtain unconditionally optimal error estimates for both the electrostatic potential and ionic concentrations of PNP equations in the $H^{1}$-norm, as well as for the velocity and pressure of NS equations in the $\mathbf{H}^{1}$- and $L^{2}$-norms, respectively. Finally, several numerical experiments are presented to support the theoretical analysis of convergence and to illustrate the satisfactory performance of the method in simulating the onset of electrokinetic instabilities in ionic fluids, and studying how they are influenced by different values of ion concentration and applied voltage. These tests relate to applications in the desalination of water.
