Optimal error estimates of a second-order temporally finite element method for electrohydrodynamic equations

In this work, we mainly present the optimal convergence rates of the temporally second-order finite element scheme for solving the electrohydrodynamic equation. Suffering from the highly coupled nonlinearity, the convergence analysis of the numerical schemes for such a system is rather rare, not to mention the optimal error estimates for the high-order temporally scheme. To this end, we abandon the traditional error analysis method following the process of energy estimate, which may lead to the loss of accuracy. Instead, we note that the charge density also possesses the "energy" decaying property directly derived by its governing equation, although it does not appear in the energy stability analysis. This fact allows us to control the error terms of the charge density more conveniently, which finally leads to the optimal convergence rates. Several numerical examples are provided to demonstrate the theoretical results, including the energy stability, mass conservation, and convergence rates.