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Parametric charge-conservative mixed finite element method for 3D incompressible inductionless MHD equations on curved domains

Xue Jiang, Lei Li, Lingxiao Li

TL;DR

This work tackles the discretization of the stationary incompressible inductionless MHD equations on curved 3D domains with a focus on charge conservation. It develops a parametric mixed finite element method that combines isoparametric Taylor-Hood elements with grad-div stabilization for $(\boldsymbol u,p)$ and parametric Brezzi-Douglas-Marini elements for the current density, using the Piola transform to enforce $\operatorname{div}\boldsymbol J=0$ exactly on the curved mesh. A rigorous a priori error analysis accounts for geometric approximation via a hold-all domain and proves optimal convergence in the energy norm and in the $L^2$-norm for the velocity, supported by a Stokes projection argument. Numerical experiments on curved meshes verify the theoretical rates and demonstrate discrete charge conservation, highlighting the method’s accuracy and robustness on curved geometries.

Abstract

This paper develops a charge-conservative mixed finite element method with optimal convergence rates for the stationary incompressible inductionless MHD equations on three-dimensional curved domains. The discretization employs the isoparametric Taylor-Hood elements with grad-div stabilization for the velocity-pressure pair, and parametric Brezzi-Douglas-Marini elements for the current density. Utilizing the Piola's transformation, the discrete current density is exactly divergence-free. By employing suitable extensions and projections, optimal a priori error estimates are derived in both the energy norm and the $L^2$-norm. Numerical experiments are presented to confirm the theoretical results.

Parametric charge-conservative mixed finite element method for 3D incompressible inductionless MHD equations on curved domains

TL;DR

This work tackles the discretization of the stationary incompressible inductionless MHD equations on curved 3D domains with a focus on charge conservation. It develops a parametric mixed finite element method that combines isoparametric Taylor-Hood elements with grad-div stabilization for and parametric Brezzi-Douglas-Marini elements for the current density, using the Piola transform to enforce exactly on the curved mesh. A rigorous a priori error analysis accounts for geometric approximation via a hold-all domain and proves optimal convergence in the energy norm and in the -norm for the velocity, supported by a Stokes projection argument. Numerical experiments on curved meshes verify the theoretical rates and demonstrate discrete charge conservation, highlighting the method’s accuracy and robustness on curved geometries.

Abstract

This paper develops a charge-conservative mixed finite element method with optimal convergence rates for the stationary incompressible inductionless MHD equations on three-dimensional curved domains. The discretization employs the isoparametric Taylor-Hood elements with grad-div stabilization for the velocity-pressure pair, and parametric Brezzi-Douglas-Marini elements for the current density. Utilizing the Piola's transformation, the discrete current density is exactly divergence-free. By employing suitable extensions and projections, optimal a priori error estimates are derived in both the energy norm and the -norm. Numerical experiments are presented to confirm the theoretical results.
Paper Structure (15 sections, 24 theorems, 159 equations, 3 figures, 3 tables)

This paper contains 15 sections, 24 theorems, 159 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Mathematical model
  3. Dimensionless MHD equations
  4. Solenoidal function spaces
  5. Continuous mixed variational formulation
  6. A charge-conservative parametric mixed finite element method
  7. Meshes and mapping
  8. Preliminaries
  9. Parametric mixed finite element approximation
  10. A priori Error estimates
  11. Preliminaries
  12. Optimal error estimate in the energy-norm
  13. Optimal error estimate in the ${\boldsymbol{L}}^2$-norm for velocity
  14. Numerical experiments
  15. Conclusion

Key Result

Lemma 2.1

The following results hold: (1) Let ${\boldsymbol{w}}, {\boldsymbol{u}}, {\boldsymbol{v}}$ in ${\boldsymbol{V}}$, we have (2) Let ${\boldsymbol{w}}, {\boldsymbol{v}}, {\boldsymbol{v}}$ in ${\boldsymbol{V}}$, we have

Figures (3)

  • Figure 1: 3D view of curved meshes $\mathcal{T}_1$ (left) and $\mathcal{T}_2$ (right).
  • Figure 2: The decreasing trend of ${\boldsymbol{L}}^2$-error of ${\boldsymbol{u}}_h$ and ${\boldsymbol{L}}^2$-error of ${\boldsymbol{J}}_h$. The symbol with $\widehat{\Omega}_h$ indicates computation on straight mesh.
  • Figure 3: The decreasing trend of energy error of ${\boldsymbol{u}}_h$ and $L^2$-error of $p_h$. The symbol with $\widehat{\Omega}_h$ indicates computation on straight mesh.

Theorems & Definitions (30)

  • Remark 1
  • Lemma 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Lemma 2.5
  • Theorem 2.6
  • Remark 2
  • Remark 3
  • Remark 4
  • ...and 20 more