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Robust globally divergence-free HDG finite element method for steady thermally coupled incompressible MHD flow

Min Zhang, Zimo Zhu, Qijia Zhai, Xiaoping Xie

TL;DR

This work develops a high-order HDG method for the steady thermally coupled incompressible MHD system, achieving globally divergence-free velocity and magnetic fields and enabling a reduced-size discrete system. The scheme uses polynomials of degree $k$ for interior and trace approximations, and introduces operators to couple interior and boundary unknowns while preserving key physical properties. The authors establish existence and, under a smallness condition, uniqueness of the discrete solution, and prove optimal $O(h^k)$ convergence for the main fields with demonstrated pressure-robustness. Numerical experiments in 2D and 3D corroborate the theoretical results, showing expected convergence rates and divergence-free behavior, highlighting the method’s suitability for robust MHD simulations.

Abstract

This paper develops an hybridizable discontinuous Galerkin (HDG) finite element method of arbitrary order for the steady thermally coupled incompressible Magnetohydrodynamics (MHD) flow. The HDG scheme uses piecewise polynomials of degrees $k(k\geq 1),k,k-1,k-1$, and $k$ respectively for the approximations of the velocity, the magnetic field, the pressure, the magnetic pseudo-pressure, and the temperature in the interior of elements, and uses piecewise polynomials of degree $k$ for their numerical traces on the interfaces of elements. The method is shown to yield globally divergence-free approximations of the velocity and magnetic fields. Existence and uniqueness results for the discrete scheme are given and optimal a priori error estimates are derived. Numerical experiments are provided to verify the obtained theoretical results.

Robust globally divergence-free HDG finite element method for steady thermally coupled incompressible MHD flow

TL;DR

This work develops a high-order HDG method for the steady thermally coupled incompressible MHD system, achieving globally divergence-free velocity and magnetic fields and enabling a reduced-size discrete system. The scheme uses polynomials of degree for interior and trace approximations, and introduces operators to couple interior and boundary unknowns while preserving key physical properties. The authors establish existence and, under a smallness condition, uniqueness of the discrete solution, and prove optimal convergence for the main fields with demonstrated pressure-robustness. Numerical experiments in 2D and 3D corroborate the theoretical results, showing expected convergence rates and divergence-free behavior, highlighting the method’s suitability for robust MHD simulations.

Abstract

This paper develops an hybridizable discontinuous Galerkin (HDG) finite element method of arbitrary order for the steady thermally coupled incompressible Magnetohydrodynamics (MHD) flow. The HDG scheme uses piecewise polynomials of degrees , and respectively for the approximations of the velocity, the magnetic field, the pressure, the magnetic pseudo-pressure, and the temperature in the interior of elements, and uses piecewise polynomials of degree for their numerical traces on the interfaces of elements. The method is shown to yield globally divergence-free approximations of the velocity and magnetic fields. Existence and uniqueness results for the discrete scheme are given and optimal a priori error estimates are derived. Numerical experiments are provided to verify the obtained theoretical results.

Paper Structure

This paper contains 13 sections, 18 theorems, 102 equations, 1 figure, 4 tables.

Table of Contents

  1. Introduction
  2. Weak problem
  3. Notation
  4. Weak form
  5. Hybridizable discontinuous Galerkin finite element method
  6. HDG scheme
  7. Preliminary results
  8. Existence and uniqueness of discrete solution
  9. Stability conditions
  10. Existence and uniqueness results
  11. Error estimates
  12. Numerical examples
  13. Conclusions

Key Result

Lemma 3.1

CX2024 For any $(\mathcal{V}_h,\mathcal{W}_h,\mathcal{Z}_h)\in[\mathbf{V}_h\times\hat{\mathbf{V}}_h^0]\times[\mathbf{V}_h\times\hat{\mathbf{W}}_h^0]\times[Z_h\times\hat{Z}_h^0]$, there hold

Figures (1)

  • Figure 1: The meshes: $(a)$$4\times4$ mesh for $\Omega=[0,1]^2$; $(b)$$4\times4\times4$ mesh for $\Omega=[0,1]^3$.

Theorems & Definitions (30)

  • Remark 2.1
  • Lemma 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • Lemma 3.4
  • Lemma 3.5
  • Lemma 3.6
  • Lemma 3.7
  • Lemma 3.8
  • ...and 20 more