Mixed finite element analysis for the Poisson-Nernst-Planck/Stokes coupling

In this paper, a type of mixed finite element method is developed to solve the Poisson-Nernst-Planck/Stokes coupling problem which is adopted to model charged fluids through the transport coupling between Stokes equations of an incompressible fluid and Poisson-Nernst-Planck (PNP) equations of a diffuse charge system. The Taylor-Hood Pk+1Pk mixed element is employed to discretize both mixed Poisson equations and Stokes equations, and the standard Pk finite element is used to discretize Nernst-Planck equations. Optimal convergence rates for both the electrostatic potential and ionic concentrations of PNP equations are obtained in both L2 and H1 norms, simultaneously, optimal convergence rates are also obtained for the velocity and pressure of Stokes equations in [H1]d and L2 norm, respectively. Numerical experiments validate the theoretical results.