Optimal error estimates of coupled and divergence-free virtual element methods for the Poisson--Nernst--Planck/Navier--Stokes equations

Abstract

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 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.Keywords Coupled Poisson-Nernst-Planck/Navier-Stokes equations • mixed virtual element method • optimal convergence • charged species transport • electrokinetic instability • water desalination.

Error Analysis of Virtual Element Methods for the Time-Dependent Poisson-Nernst-Planck Equations

“…Recently, we note that a fully coupled and energy-stable VEM was proposed and analyzed for the coupled PNP and Navier-Stokes equations in [21], (the preprint of which was submitted to arXiv at a similar time as ours, see [20,50]). The existence and uniqueness of the virtual element approximation are presented and the optimal error estimates are derived for fully discrete scheme in the L 2 and H 1 norms, respectively.…”

Direct numerical simulation of electrokinetic transport phenomena: variational multi-scale stabilization and octree-based mesh refinement

“…We build upon our previous work on a finite elements based framework for solving the NS and PNP equations [30]. Finite element approaches have been successfully applied for high-resolution simulations of NS (See [31] for an overview) and PNP [32][33][34][35][36][37][38][39][40] equations. They are well suited for the NS-PNP system of equations due to (a) the use of a variational formulation that allows the weakening of derivatives, (b) the natural incorporation of heterogeneous and mixed boundary conditions often seen in these systems, and (c) ability to construct rigorous a posteriori error estimates for mesh adaptivity that enable substantial computational gains away from the EDLs.…”