Optimal Error Estimates of Coupled and Divergence-Free Virtual Element Methods for the Poisson–Nernst–Planck/Navier–Stokes Equations and Applications in Electrochemical Systems

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. These equations model microfluidic and electrochemical systems that include the 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. The global mass conservation and electric energy decay of the scheme are also established. Also, we rigorously derive unconditionally optimal error estimates for both the electrostatic potential and ionic concentrations of PNP equations in the L^2 and H^1 -norms, as well as for the velocity and pressure of NS equations in the L^2 , 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 are relevant in applications of water desalination.