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Optimal ${L^2}$ error estimates of fully discrete finite element methods for the 2D/3D diffuse interface two-phase MHD flows

Ke Zhang, Haiyan Su

Abstract

In this paper, we perform an optimal L2-norm error analysis of a fully discrete convex-splitting finite element method (FEM) for the two-phase diffuse interface magnetohydrodynamics (MHD) system. The method use the semi-implicit backward Euler scheme in time and use the standard inf-sup stable Taylor-Hood/Mini elements to discretize the velocity and pressure. The previous works provided the optimal H1-norm error estimates for all components, but not the optimal L2-norm estimates, which are caused by the nonlinear coupled terms. The optimal L2-norm error analysis is achieved through the novel Ritz and Stokes quasi-projections. In addition, the mass conservation and unconditional energy stability of the finite element convex-splitting scheme are ensured. Numerical examples are presented to validate the theoretical analysis.

Optimal ${L^2}$ error estimates of fully discrete finite element methods for the 2D/3D diffuse interface two-phase MHD flows

Abstract

In this paper, we perform an optimal L2-norm error analysis of a fully discrete convex-splitting finite element method (FEM) for the two-phase diffuse interface magnetohydrodynamics (MHD) system. The method use the semi-implicit backward Euler scheme in time and use the standard inf-sup stable Taylor-Hood/Mini elements to discretize the velocity and pressure. The previous works provided the optimal H1-norm error estimates for all components, but not the optimal L2-norm estimates, which are caused by the nonlinear coupled terms. The optimal L2-norm error analysis is achieved through the novel Ritz and Stokes quasi-projections. In addition, the mass conservation and unconditional energy stability of the finite element convex-splitting scheme are ensured. Numerical examples are presented to validate the theoretical analysis.

Paper Structure

This paper contains 15 sections, 6 theorems, 74 equations, 7 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Main results
  3. Preliminaries and weak formulation
  4. Numerical scheme and main results
  5. Projections and their properties
  6. The Proof of Theorem \ref{['theorem2-1']}
  7. Estimates for $e_{\textbf{u}}^{k+1}$.
  8. Estimates for $e_{\textbf{B}}^{k+1}$.
  9. Numerical examples
  10. 2D/3D convergence of the scheme
  11. Spinodal decomposition
  12. Two-phase Kelvin-Helmholtz instability problem
  13. Dynamics of single mode sinusoidal perturbation
  14. Dynamics of double mode sinusoidal perturbation
  15. Conclusion Remarks

Key Result

Lemma 2.1

Based on the $Poincar\acute{e}$ inequalities and embedding inequalities in 1975SobolevJean2006Mathematical1986On2019A, we have where $C_{0}$ denotes a generic positive constant independent of $\Delta t$, $h$, and discretization parameters, which may take different values at different places.

Figures (7)

  • Figure 5.1: Snapshots of phase field dynamical evolution for spinodal decomposition for case I.
  • Figure 5.2: The system energy (left), algorithm energy (middle) and the discrete mass (right) for case I.
  • Figure 5.3: Snapshots of the phase field perturbed sinusoidal at $t=0.001$ (a), 0.6 (b), 0.85 (c), 1 (d), 1.1 (e), 1.2 (f), 1.4 (g), 1.6 (h) for case I.
  • Figure 5.4: Snapshots of the vorticity dynamics at time at $t=0.001$ (a), 0.6 (b), 0.85 (c), 1 (d), 1.1 (e), 1.2 (f), 1.4 (g), 1.6 (h) for case I.
  • Figure 5.5: Snapshots of the phase field (upper), vorticity dynamics (lower) perturbed sinusoidal at $t=0.001$ (a), 0.3 (b), 0.5 (c), 0.75 (d) for case I.
  • ...and 2 more figures

Theorems & Definitions (9)

  • Definition 2.1
  • Remark 2.1
  • Lemma 2.1
  • Proposition 2.1
  • Theorem 2.1
  • Remark 3.1
  • Lemma 4.1
  • Lemma 4.2
  • Lemma 4.3