Stable finite element methods preserving ∇ · B= 0 exactly for MHD models

Kaibo Hu, Yicong Ma, Jinchao Xu

Research output: Contribution to journalArticle

17 Citations (Scopus)

Abstract

This paper is devoted to the design and analysis of some structure-preserving finite element schemes for the magnetohydrodynamics (MHD) system. The main feature of the method is that it naturally preserves the important Gauss’s law, namely ∇ · B= 0. In contrast to most existing approaches that eliminate the electrical field variable E and give a direct discretization of the magnetic field, our new approach discretizes the electric field E by Nédélec type edge elements for H(curl) , while the magnetic field B by Raviart–Thomas type face elements for H(div). As a result, the divergence-free condition on the magnetic field holds exactly on the discrete level. For this new finite element method, an energy stability estimate can be naturally established in an analogous way as in the continuous case. Furthermore, well-posedness is rigorously established in the paper for the Picard linearization of the fully nonlinear systems by using the Brezzi theory. This well-posedness naturally leads to robust (and optimal) preconditioners for the linearized systems.

Original languageEnglish (US)
Pages (from-to)371-396
Number of pages26
JournalNumerische Mathematik
Volume135
Issue number2
DOIs
StatePublished - Feb 1 2017

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Magnetohydrodynamics
Finite Element Method
Magnetic Field
Magnetic fields
Finite element method
Well-posedness
Edge Elements
Divergence-free
Stability Estimates
Curl
Energy Estimates
Fully Nonlinear
Convergence of numerical methods
Linearization
Preconditioner
Nonlinear systems
Electric Field
Eliminate
Nonlinear Systems
Discretization

All Science Journal Classification (ASJC) codes

  • Computational Mathematics
  • Applied Mathematics

Cite this

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Stable finite element methods preserving ∇ · B= 0 exactly for MHD models. / Hu, Kaibo; Ma, Yicong; Xu, Jinchao.

In: Numerische Mathematik, Vol. 135, No. 2, 01.02.2017, p. 371-396.

Research output: Contribution to journalArticle

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