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Analysis of an Unconditionally Convergent Stabilized Finite Element Formulation for Incompressible Magnetohydrodynamics

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Abstract

In this work, we analyze a recently proposed stabilized finite element formulation for the approximation of the resistive magnetohydrodynamics equations. The novelty of this formulation with respect to existing ones is the fact that it always converges to the physical solution, even when it is singular. We have performed a detailed stability and convergence analysis of the formulation in a simplified setting. From the convergence analysis, we infer that a particular type of meshes with a macro-element structure is needed, which can be easily obtained after a straight modification of any original mesh.

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Notes

  1. In any case, we can always project the continuous fields into the FE spaces using proper projections.

  2. We note that the continuous problem is singular for \(\mathrm{Rm} = \infty \). High magnetic Reynolds numbers appear in astrophysical simulations. However, these simulations always involve transient systems, i.e. the ideal MHD system (\(\mathrm{Rm} = \infty \)) cannot be stated in steady form. Even though the numerical analysis in this work has been restricted to the steady case for simplicity, the method has been conceived and numerically tested for the transient problem [7]. The first value in \(\tau _3\) does not blow up for \(\mathrm{Rm} = \infty \) when applied to transient problem, since it will include a time-step size \(\delta t\) dependency when using a quasi-static approach or a dynamic subgrid stabilization will be used (see [23] for details).

  3. The analysis for non-degenerate meshes and variable stabilization parameters introduces some additional technicalities. We refer to [4, 20] for the numerical analysis of stabilized FE discretizations of Navier–Stokes and Maxwell problems respectively, in this more general setting.

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Authors and Affiliations

  1. Centre Internacional de Metodes Numerics en Enginyeria (CIMNE), Universitat Politècnica de Catalunya (UPC), Castelldefels, Spain

    Santiago Badia & Ramon Planas

  2. Parc Mediterrani de la Tecnologia, UPC, Esteve Terradas 5, 08860, Castelldefels, Spain

    Santiago Badia & Ramon Planas

  3. Universitat Politècnica de Catalunya (UPC), Campus Nord, Jordi Girona 1-3, Edifici C1, 08034, Barcelona, Spain

    Ramon Codina

Authors
  1. Santiago Badia
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  2. Ramon Codina
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  3. Ramon Planas
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Corresponding author

Correspondence to Ramon Planas.

Additional information

The work of the first and third authors was funded by the European Research Council under the FP7 Programme Ideas through the Starting Grant No. 258443—COMFUS: Computational Methods for Fusion Technology and the project FUSSIM, Ref. ENE2011-28556, from the Spanish Government. The second author has been partially supported by the Consolider-Ingenio project TECNOFUS, Ref. CSD2008-00079, from the Spanish Ministry of Science and Innovation, and from the ICREA Acadèmia Program, from the Catalan Government. Finally, the third author would like to acknowledge the support received from the Universitat Politècnica de Catalunya (UPC) and from the Col.legi d’Enginyers de Camins, Canals i Ports de Catalunya.

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Badia, S., Codina, R. & Planas, R. Analysis of an Unconditionally Convergent Stabilized Finite Element Formulation for Incompressible Magnetohydrodynamics. Arch Computat Methods Eng 22, 621–636 (2015). https://doi.org/10.1007/s11831-014-9129-5

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