Mixed Finite Element approximation of an MHD problem involving conducting and insulating regions: the 2D case

Jean Luc Guermond; Peter D. Minev

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 36, Issue: 3, page 517-536
  • ISSN: 0764-583X

Abstract

top
We show that the Maxwell equations in the low frequency limit, in a domain composed of insulating and conducting regions, has a saddle point structure, where the electric field in the insulating region is the Lagrange multiplier that enforces the curl-free constraint on the magnetic field. We propose a mixed finite element technique for solving this problem, and we show that, under mild regularity assumption on the data, Lagrange finite elements can be used as an alternative to edge elements.

How to cite

top

Guermond, Jean Luc, and Minev, Peter D.. "Mixed Finite Element approximation of an MHD problem involving conducting and insulating regions: the 2D case." ESAIM: Mathematical Modelling and Numerical Analysis 36.3 (2010): 517-536. <http://eudml.org/doc/194115>.

@article{Guermond2010,
abstract = { We show that the Maxwell equations in the low frequency limit, in a domain composed of insulating and conducting regions, has a saddle point structure, where the electric field in the insulating region is the Lagrange multiplier that enforces the curl-free constraint on the magnetic field. We propose a mixed finite element technique for solving this problem, and we show that, under mild regularity assumption on the data, Lagrange finite elements can be used as an alternative to edge elements. },
author = {Guermond, Jean Luc, Minev, Peter D.},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Finite element method; Magnetohydrodynamics.; finite element method; magnetohydrodynamics},
language = {eng},
month = {3},
number = {3},
pages = {517-536},
publisher = {EDP Sciences},
title = {Mixed Finite Element approximation of an MHD problem involving conducting and insulating regions: the 2D case},
url = {http://eudml.org/doc/194115},
volume = {36},
year = {2010},
}

TY - JOUR
AU - Guermond, Jean Luc
AU - Minev, Peter D.
TI - Mixed Finite Element approximation of an MHD problem involving conducting and insulating regions: the 2D case
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 36
IS - 3
SP - 517
EP - 536
AB - We show that the Maxwell equations in the low frequency limit, in a domain composed of insulating and conducting regions, has a saddle point structure, where the electric field in the insulating region is the Lagrange multiplier that enforces the curl-free constraint on the magnetic field. We propose a mixed finite element technique for solving this problem, and we show that, under mild regularity assumption on the data, Lagrange finite elements can be used as an alternative to edge elements.
LA - eng
KW - Finite element method; Magnetohydrodynamics.; finite element method; magnetohydrodynamics
UR - http://eudml.org/doc/194115
ER -

References

top
  1. T. Amari, J.F. Luciani and P. Joly, A preconditioned semi-implicit method for magnetohydrodynamics equation. SIAM J. Sci. Comput.21 (1999) 970-986.  
  2. F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer Verlag, New York, Springer Ser. Comput. Math. 15 (1991).  
  3. A. Bossavit, Electromagnétisme en vue de la modélisation. SMAI/Springer-Verlag, Paris, Math. Appl. 14 (1993). See also Computational Electromagnetism, Variational Formulations, Complementary, Edge Elements, Academic Press (1998).  
  4. H. Brezis, Analyse fonctionnelle. Masson, Paris (1991).  
  5. P. Clément, Approximation by finite element functions using local regularization. Anal. Numér.9 (1975) 77-84.  
  6. M. Costabel, A coercive bilinear form for Maxwell's equations. J. Math. Anal. Appl.157 (1991) 527-541.  
  7. M.L. Dudley and R.W. James, time-dependent kinematic dynamos with stationary flows. Proc. Roy. Soc. LondonA425 (1989) 407-429.  
  8. L. Demkowicz and L. Vardapetyan, Modeling of electromagnetic absorption/scattering problems using hp-adaptive finite elements. Comput. Methods Appl. Mech. Engrg.152 (1998) 103-124. Symposium on Advances in Computational Mechanics, Vol. 5 (Austin, TX, 1997).  
  9. J.-F. Gerbeau, A stabilized finite element method for the incompressible magnetohydrodynamic equations. Numer. Math.87 (2000) 83-111.  
  10. J.-L. Guermond, J. Léorat and C. Nore, Numerical simulations of 2D MHD problems using Lagrange finite elements (in preparation 2001).  
  11. J.-L. Guermond and P.D. Minev, Mixed finite element approximation of an MHD problem involving conducting and insulating regions: the 3D case (submitted 2002).  
  12. V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations. Springer-Verlag, Berlin, Springer Ser. Comput. Math. 5 (1986).  
  13. J. Léorat, Numerical simulations of cylindrical dynamos: scope and method. In 7th beer-Sheva Onternatal seminar, Vol. 162, pp. 282-292. AIAA Progress in Astronautics and aeronautic series, 1994.  
  14. J. Léorat, Linear dynamo simulations with time dependent helical flows. Magnetohydrodynamics31 (1995) 367-373.  
  15. J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications, Vol. 1. Dunod, Paris (1968).  
  16. H.K. Moffatt, Magnetic Field Generation in Electrically Conducting Fluids. Cambridge Monographs on Mechanics and Applied Mathematics. Cambridge University Press, Cambridge (1978).  
  17. A.J. Meir and P.G. Schmidt, Analysis and numerical approximation of a stationary MHD flow problem with non-ideal boundary. SIAM J. Numer. Anal.36 (1999) 1304-1332.  
  18. J. Necas, Les méthodes directes en théorie des équations elliptiques. Masson, Paris (1967).  
  19. J.-C. Nédélec, A new family of mixed finite elements in ℝ 3 . Numer. Math.50 (1986) 57-81.  
  20. R.L. Parker, Reconnexion of lines of force in rotating spheres and cylinders. Proc. Roy. Soc.291 (1966) 60-72.  
  21. N. Ben Salah, A. Soulaimani and W.G. Habashi, A finite element method for magnetohydrodynamics. Comput. Methods Appl. Mech. Engrg.190 (2001) 5867-5892.  
  22. N. Ben Salah, A. Soulaimani, W.G. Habashi and M. Fortin, A conservative stabilized finite element method for magnetohydrodynamics equations. Internat. J. Numer. Methods Fluids29 (1999) 535-554.  
  23. R. Verfürth, Error estimates for a mixed finite element approximation of the Stokes equation. RAIRO Anal. Numér.18 (1984) 175-182.  

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.