An efficient unconditional energy-stable finite element method for the electro-hydrodynamic equations

In this paper, we mainly focus on the numerical approximations of the electro-hydrodynamic system, which couples the Poisson-Nernst-Planck equations and the Navier-Stokes equations. A novel linear, fully-decoupled and energy-stable finite element scheme for solving this system is proposed and analyzed. The fully discrete scheme developed here is employed by the stabilizing strategy, implicit-explicit (IMEX) scheme and a rotational pressure-correction method. One particular feature of the scheme is adding a stabilization term artificially in the conservation of charge density equation to decouple the computations of velocity field from electric field, which can be treated as a first-order perturbation term for balancing the explicit treatment of the coupling term. We rigorously prove the unique solvability, unconditional energy stability and error estimates of the proposed scheme. Finally, some numerical examples are provided to verify the accuracy and stability of the developed numerical scheme.