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, energy-stable virtual element method (VEM) for solving the coupled Poisson-Nernst-Planck (PNP) Navier--Stokes (NS) equations modeling microfluidic electrochemical systems (diffuse transport of charged species within incompressible fluids through electrostatic forces). A mixed VEM is employed to discretize NS whereas classical in primal form used PNP equations. The stability, existence uniqueness solution associated are proved by fixed point theory. Global mass conservation electric energy decay scheme also proved. Also, obtain unconditionally optimal error estimates both potential ionic concentrations $H^{1}$-norm, as well velocity pressure $\mathbf{H}^{1}$- $L^{2}$-norms, respectively. Finally, several numerical experiments presented support theoretical analysis convergence illustrate satisfactory performance simulating onset electrokinetic instabilities fluids, studying how they influenced different values ion concentration applied voltage. These tests relate applications desalination water.