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Robust Finite Elements for linearized Magnetohydrodynamics

L. Beirão da Veiga, F. Dassi, G. Vacca

TL;DR

This work develops a pressure-robust finite element method for the linearized three-dimensional MHD equations, achieving quasi-robustness in high advection and magnetic-diffusion regimes. The method couples a divergence-free nonconforming $\mathrm{BDM}_k$ velocity discretization with an $H^1$-conforming magnetic discretization and adds a CIP-type stabilization for the fluid–magnetic interaction, along with an upwind DG treatment for convection. A rigorous stability and error framework is established, including convergence without solution regularity and clear quasi-robustness with respect to diffusion parameters, complemented by numerical experiments that confirm optimal convergence rates in both diffusive and convective scenarios and demonstrate robustness to small diffusivities. The results are relevant for efficiently solving linearized MHD problems and provide a solid foundation for extending the approach to time-dependent and nonlinear regimes.

Abstract

We introduce a pressure robust Finite Element Method for the linearized Magnetohydrodynamics equations in three space dimensions, which is provably quasi-robust also in the presence of high fluid and magnetic Reynolds numbers. The proposed scheme uses a non-conforming BDM approach with suitable DG terms for the fluid part, combined with an $H^1$-conforming choice for the magnetic fluxes. The method introduces also a specific CIP-type stabilization associated to the coupling terms. Finally, the theoretical result are further validated by numerical experiments.

Robust Finite Elements for linearized Magnetohydrodynamics

TL;DR

This work develops a pressure-robust finite element method for the linearized three-dimensional MHD equations, achieving quasi-robustness in high advection and magnetic-diffusion regimes. The method couples a divergence-free nonconforming velocity discretization with an -conforming magnetic discretization and adds a CIP-type stabilization for the fluid–magnetic interaction, along with an upwind DG treatment for convection. A rigorous stability and error framework is established, including convergence without solution regularity and clear quasi-robustness with respect to diffusion parameters, complemented by numerical experiments that confirm optimal convergence rates in both diffusive and convective scenarios and demonstrate robustness to small diffusivities. The results are relevant for efficiently solving linearized MHD problems and provide a solid foundation for extending the approach to time-dependent and nonlinear regimes.

Abstract

We introduce a pressure robust Finite Element Method for the linearized Magnetohydrodynamics equations in three space dimensions, which is provably quasi-robust also in the presence of high fluid and magnetic Reynolds numbers. The proposed scheme uses a non-conforming BDM approach with suitable DG terms for the fluid part, combined with an -conforming choice for the magnetic fluxes. The method introduces also a specific CIP-type stabilization associated to the coupling terms. Finally, the theoretical result are further validated by numerical experiments.
Paper Structure (14 sections, 15 theorems, 131 equations, 8 figures)

This paper contains 14 sections, 15 theorems, 131 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Continuous problem
  3. Notations and preliminary theoretical results
  4. Stabilized finite element method
  5. Discrete spaces and interpolation analysis
  6. Discrete forms
  7. Discrete scheme
  8. Theoretical analysis
  9. Stability analysis
  10. Error analysis
  11. Convergence without requirements on solution regularity
  12. Numerical Experiments
  13. Fluid and magneto convective dominant regime
  14. Magneto convective dominant regime

Key Result

Proposition 2.1

Assume that the domain $\Omega$ is a convex polyhedron. Then Problem eq:linear variazionale is well-posed. Additionally, Problem eq:linear variazionale is a variational formulation of Problem eq:linear primale.

Figures (8)

  • Figure 1: mesh 2 and a detail inside.
  • Figure 2: Convection dominant regime with different values of $\nu_{\rm S}$ and $\nu_{\rm M}$, $k=1$.
  • Figure 3: Convection dominant regime with different values of $\nu_{\rm S}$ and $\nu_{\rm M}$, $k=2$.
  • Figure 4: Stability with respect to $\nu_{\rm S}$ and $\nu_{\rm M}$ for $k=1$ and 2.
  • Figure 5: Magneto convective dominant regime $\nu_{\rm S}=\texttt{1e-10}$ and $\nu_{\rm M}=\texttt{1e-02}$, $k=1$.
  • ...and 3 more figures

Theorems & Definitions (35)

  • Proposition 2.1
  • proof
  • Remark 2.1
  • Remark 3.1
  • Lemma 3.2: Trace inequality
  • Lemma 3.3: Bramble-Hilbert
  • Lemma 3.4: Inverse estimate
  • Remark 3.5
  • Lemma 4.1: Interpolation operator on $\boldsymbol{V}^h_k$
  • Remark 4.2
  • ...and 25 more