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

Shengfeng Wang, Zeyu Xia, Maojun Li

TL;DR

This work addresses the challenge of obtaining sharp error estimates for a second-order temporally discretized electrohydrodynamics (EHD) system that couples incompressible Navier–Stokes with electrostatic effects. By leveraging the energy-decay property of the charge density and a carefully crafted fully discrete scheme using the second-order backward differentiation formula (BDF2) with Taylor–Hood finite elements, the authors prove unconditional energy stability, discrete charge conservation, and unique solvability. The main theoretical contribution is an optimal error bound of the form $\mathcal{O}(\\tau^2 + h^{r+1})$ in the $L^2$-norm for the potential, charge density, and velocity, established through projections, error equations, induction, and a duality argument for the potential. Numerical experiments validate the convergence rates, energy stability, and mass conservation, confirming the practical reliability and accuracy of the proposed method for EHD simulations.

Abstract

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.

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

TL;DR

This work addresses the challenge of obtaining sharp error estimates for a second-order temporally discretized electrohydrodynamics (EHD) system that couples incompressible Navier–Stokes with electrostatic effects. By leveraging the energy-decay property of the charge density and a carefully crafted fully discrete scheme using the second-order backward differentiation formula (BDF2) with Taylor–Hood finite elements, the authors prove unconditional energy stability, discrete charge conservation, and unique solvability. The main theoretical contribution is an optimal error bound of the form in the -norm for the potential, charge density, and velocity, established through projections, error equations, induction, and a duality argument for the potential. Numerical experiments validate the convergence rates, energy stability, and mass conservation, confirming the practical reliability and accuracy of the proposed method for EHD simulations.

Abstract

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.
Paper Structure (14 sections, 5 theorems, 101 equations, 1 figure, 2 tables)

This paper contains 14 sections, 5 theorems, 101 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Main results
  3. Variational formulation
  4. Numerical schemes
  5. Energy stability and unique solvability
  6. Optimal error estimates
  7. Preliminaries
  8. Error equations
  9. Estimates for $\hbox{\boldmath $n$}\,{\bf =0}$
  10. Estimates for $\hbox{\boldmath $n$}\,{\bf \geq1}$
  11. Numerical Examples
  12. Convergence test
  13. Stability and mass conservation
  14. Conclusion

Key Result

Theorem 2.1

The variational formulation var1-var4 is energy-stable and satisfies the following energy inequality

Figures (1)

  • Figure 1: Discrete energy stability and charge mass conservation at each time level

Theorems & Definitions (7)

  • Theorem 2.1
  • proof
  • Theorem 2.2
  • Theorem 2.3
  • proof
  • Lemma 3.1: Brenner2008Girault1986Thomee2006Wheeler1973
  • Lemma 3.2: Brenner2008