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A Reynolds- and Hartmann-semirobust hybrid method for magnetohydrodynamics

Daniele A. Di Pietro, Jerome Droniou, Vito Patierno

Abstract

We propose and analyze a new method for the unsteady incompressible magnetohydrodynamics equations on convex domains with hybrid approximations of both vector-valued and scalar-valued fields. The proposed method is convection-semirobust, meaning that, for sufficiently smooth solutions, one can derive a priori estimates for the velocity and the magnetic field that do not depend on the inverse of the diffusion coefficients. This is achieved while at the same time providing relevant additional features, namely an improved order of convergence for the (asymptotic) diffusion-dominated regime, a small stencil (owing to the absence of inter-element penalty terms), and the possibility to significantly reduce the size of the algebraic problems through static condensation. The theoretical results are confirmed by a complete panel of numerical experiments.

A Reynolds- and Hartmann-semirobust hybrid method for magnetohydrodynamics

Abstract

We propose and analyze a new method for the unsteady incompressible magnetohydrodynamics equations on convex domains with hybrid approximations of both vector-valued and scalar-valued fields. The proposed method is convection-semirobust, meaning that, for sufficiently smooth solutions, one can derive a priori estimates for the velocity and the magnetic field that do not depend on the inverse of the diffusion coefficients. This is achieved while at the same time providing relevant additional features, namely an improved order of convergence for the (asymptotic) diffusion-dominated regime, a small stencil (owing to the absence of inter-element penalty terms), and the possibility to significantly reduce the size of the algebraic problems through static condensation. The theoretical results are confirmed by a complete panel of numerical experiments.
Paper Structure (21 sections, 7 theorems, 91 equations, 2 figures)

This paper contains 21 sections, 7 theorems, 91 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Continuous problem
  3. Discrete setting
  4. Mesh
  5. Polynomial spaces
  6. Discrete spaces
  7. Discrete inner products and norms
  8. Discrete forms
  9. Diffusion
  10. Solenoidal coupling
  11. Convection
  12. Discrete problem and well-posedness
  13. Discrete problem
  14. Energy norm and well-posedness
  15. Error analysis
  16. ...and 6 more sections

Key Result

Lemma 3

For all $v\in H^1(\Omega)^d$, it holds

Figures (2)

  • Figure 1: Error ${\vert\vert\vert (\underline{e}_{u,h},\underline{e}_{b,h}) \vert\vert\vert}_h$ vs. meshsize $h$ for the two-dimensional test case of Section \ref{['sec:numerical.results:2d']}.
  • Figure 2: Error ${\vert\vert\vert (\underline{e}_{u,h},\underline{e}_{b,h}) \vert\vert\vert}_h$ vs. meshsize $h$ for the three-dimensional test case of Section \ref{['sec:numerical.results:3d']}.

Theorems & Definitions (19)

  • Remark 1: Space and boundary conditions for the magnetic field
  • Remark 2: Validity of \ref{['eq:modified']} in two space dimensions
  • Lemma 3: $L^2$-boundedness of the velocity interpolator
  • proof
  • Remark 4: Pointwise divergence-free vector fields
  • Remark 5: Interpolates of divergence-free functions
  • Lemma 6: Inf-sup condition on $B_h$
  • Lemma 7: Skew-symmetry of $t_{h}$
  • proof
  • Theorem 8: Well-posedness of the scheme
  • ...and 9 more