Convergence analysis of a fully discrete finite element method for the EHD system

This paper develops and analyzes a fully discrete finite element method for the electrohydrodynamic (EHD) model in three dimensions. The arising nonlinear system couples the incompressible Navier-Stokes equations for velocity and pressure with a convection diffusion equation for the charge density and a Poisson equation for the electric potential through the convection term, electro-migration term, Coulomb law and electrical force. A fully discrete scheme, which is based on the finite element method for the spatial discretization and first order semi-implicit scheme for the temporal discretization, is proposed to solve this multiphysics coupled system. It is shown that the proposed scheme is uniquely solvable and satisfies a total charge conservation law, and a discrete energy law unconditionally. The convergence of subsequences of the numerical solutions is established without extra assumptions on the regularity of the exact solution by utilizing the stability of the numerical scheme and the compactness method. As a by-product, the convergence result also provides a constructive proof of the existence of weak solution to the EHD model. Furthermore, given more regularity on the weak solution, the uniqueness of weak solution and convergence of the numerical scheme without extracting subsequences are also rigorously derived. Numerical experiments are also presented to validate the theoretical findings and to show the effectiveness of the proposed scheme.