Structure-preserving finite element methods for stationary MHD models

In this paper, we develop a class of mixed finite element scheme for stationary magnetohydrodynamics (MHD) models, using magnetic field $\bm B$ and current density $\bm j$ as the discretization variables. We show that the Gauss's law for the magnetic field, namely $\nabla\cdot\bm{B}=0$, and the energy law for the entire system are exactly preserved in the finite element schemes. Based on some new basic estimates for $H^{h}(\mathrm{div})$, we show that the new finite element scheme is well-posed. Furthermore, we show the existence of solutions to the nonlinear problems and the convergence of Picard iterations and finite element methods under some conditions.

[1]  Ming-Jiu Ni,et al.  A consistent and conservative scheme for incompressible MHD flows at a low magnetic Reynolds number. Part III: On a staggered mesh , 2012, J. Comput. Phys..

[2]  Vivette Girault,et al.  Finite Element Methods for Navier-Stokes Equations - Theory and Algorithms , 1986, Springer Series in Computational Mathematics.

[3]  Jinchao Xu,et al.  Stable finite element methods preserving $$\nabla \cdot \varvec{B}=0$$∇·B=0 exactly for MHD models , 2017, Numerische Mathematik.

[4]  D. Arnold,et al.  Finite element exterior calculus, homological techniques, and applications , 2006, Acta Numerica.

[5]  Paul R. Woodward,et al.  On the Divergence-free Condition and Conservation Laws in Numerical Simulations for Supersonic Magnetohydrodynamical Flows , 1998 .

[6]  J. Brackbill,et al.  The Effect of Nonzero ∇ · B on the numerical solution of the magnetohydrodynamic equations☆ , 1980 .

[7]  Thomas Rylander,et al.  Computational Electromagnetics , 2005, Electronics, Power Electronics, Optoelectronics, Microwaves, Electromagnetics, and Radar.

[8]  Jinchao Xu,et al.  Robust preconditioners for incompressible MHD models , 2015, J. Comput. Phys..

[9]  D. Arnold,et al.  Finite element exterior calculus: From hodge theory to numerical stability , 2009, 0906.4325.

[10]  Jinchao Xu,et al.  Stable finite element methods preserving ∇·B=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nabla \cdot \varvec{B}= , 2014, Numerische Mathematik.

[11]  Ming Wang,et al.  Convergence Analysis of Triangular MAC Schemes for Two Dimensional Stokes Equations , 2015, J. Sci. Comput..

[12]  M. Fortin,et al.  Mixed Finite Element Methods and Applications , 2013 .

[13]  Ramakanth Munipalli,et al.  A current density conservative scheme for incompressible MHD flows at a low magnetic Reynolds number. Part I: On a rectangular collocated grid system , 2007, J. Comput. Phys..

[14]  Richard S. Falk,et al.  Local bounded cochain projections , 2014, Math. Comput..

[15]  A. Bossavit.  Computational Electromagnetism: Variational Formulations, Complementarity, Edge Elements , 1997 .

[16]  M. Gunzburger,et al.  On the existence, uniqueness, and finite element approximation of solutions of the equations of stationary, incompressible magnetohydrodynamics , 1991 .

[17]  L. R. Scott,et al.  The Mathematical Theory of Finite Element Methods , 1994 .

[18]  Dominik Schötzau,et al.  Mixed finite element methods for stationary incompressible magneto–hydrodynamics , 2004, Numerische Mathematik.

[19]  M. Ni,et al.  Numerical study of MHD pressure drop in rectangular ducts with insulating coatings , 2010 .

[20]  R. Hiptmair.  Finite elements in computational electromagnetism , 2002, Acta Numerica.

[21]  F. Brezzi.  On the existence, uniqueness and approximation of saddle-point problems arising from lagrangian multipliers , 1974 .

[22]  Hans Z. Munthe-Kaas,et al.  Topics in structure-preserving discretization* , 2011, Acta Numerica.