Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Conservative nonconforming virtual element method for stationary incompressible magnetohydrodynamics

Xiaojing Dong, Yunqing Huang, Tianwen Wang

Abstract

In this paper, we propose a conservative nonconforming virtual element method for the full stationary incompressible magnetohydrodynamics model. We leverage the virtual element satisfactory divergence-free property to ensure mass conservation for the velocity field. The condition of the well-posedness of the proposed method, as well as the stability are derived. We establish optimal error estimates in the discrete energy norm for both the velocity and magnetic field. Furthermore, by employing a new technique, we obtain the optimal error estimates in $L^2$-norm without any additional conditions. Finally, numerical experiments are presented to validate the theoretical analysis. In the implementation process, we adopt the effective Oseen iteration to handle the nonlinear system.

Conservative nonconforming virtual element method for stationary incompressible magnetohydrodynamics

Abstract

In this paper, we propose a conservative nonconforming virtual element method for the full stationary incompressible magnetohydrodynamics model. We leverage the virtual element satisfactory divergence-free property to ensure mass conservation for the velocity field. The condition of the well-posedness of the proposed method, as well as the stability are derived. We establish optimal error estimates in the discrete energy norm for both the velocity and magnetic field. Furthermore, by employing a new technique, we obtain the optimal error estimates in -norm without any additional conditions. Finally, numerical experiments are presented to validate the theoretical analysis. In the implementation process, we adopt the effective Oseen iteration to handle the nonlinear system.

Paper Structure

This paper contains 9 sections, 12 theorems, 104 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Virtual element method
  4. Meshes and projections
  5. The nonconforming element space and piecewise polynomials space
  6. The nodal space
  7. The discrete formulation
  8. Error analysis
  9. Numerical examples

Key Result

Theorem 2.1

(Dong-Three) If $\frac{\sqrt[]{2} \lambda_0^2 \max\{1,\sqrt[]{2} S_c\}\|F\|_{\ast}}{(\min\{R_{\nu}^{-1},\lambda_1 R_m^{-1}S_c\})^2} < 1$, the problem (MHDrewriteeq) has a unique solution $(\boldsymbol{u},\boldsymbol{b},p) \in \boldsymbol{H}_0^1(\Omega) \times \boldsymbol{H}_n^1(\Omega) \times L_0^2(

Figures (4)

  • Figure 1: Sample meshes: $\mathcal{T}_h^1$(left), $\mathcal{T}_h^2$(middle) and $\mathcal{T}_h^3$(right).
  • Figure 2: Errors of velocity and magnetic field on meshes $\mathcal{T}_h^3$.
  • Figure 3: Errors of velocity and magnetic field on meshes $\mathcal{T}_h^1$.
  • Figure 4: Numerical (scatter), analytical (line) along $x_1 = 3$.

Theorems & Definitions (21)

  • Theorem 2.1
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • proof
  • Lemma 3.6
  • ...and 11 more