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

Abstract

Abstract 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 P k + 1 P k mixed element is employed to discretize both mixed Poisson equations and Stokes equations, and the standard P k 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 L 2 and H 1 norms, simultaneously, optimal convergence rates are also obtained for the velocity and pressure of Stokes equations in [ H 1 ] d and L 2 norm, respectively. Numerical experiments validate the theoretical results.