Mixed finite element approximation of an MHD problem involving conducting and insulating regions : the 2D case
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 36 (2002) no. 3, pp. 517-536.

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.

DOI : https://doi.org/10.1051/m2an:2002024
Classification : 65N35,  65N25,  65F05,  35J05
Mots clés : finite element method, magnetohydrodynamics
@article{M2AN_2002__36_3_517_0,
author = {Guermond, Jean Luc and Minev, Peter D.},
title = {Mixed finite element approximation of an MHD problem involving conducting and insulating regions : the 2D case},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
pages = {517--536},
publisher = {EDP-Sciences},
volume = {36},
number = {3},
year = {2002},
doi = {10.1051/m2an:2002024},
zbl = {1137.65437},
mrnumber = {1918943},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an:2002024/}
}
Guermond, Jean Luc; 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 - Modélisation Mathématique et Analyse Numérique, Tome 36 (2002) no. 3, pp. 517-536. doi : 10.1051/m2an:2002024. http://www.numdam.org/articles/10.1051/m2an:2002024/

[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. | Zbl 0964.76057

[2] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer Verlag, New York, Springer Ser. Comput. Math. 15 (1991). | MR 1115205 | Zbl 0788.73002

[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). | MR 1488417

[4] H. Brezis, Analyse fonctionnelle. Masson, Paris (1991). | MR 697382 | Zbl 0511.46001

[5] P. Clément, Approximation by finite element functions using local regularization. Anal. Numér. 9 (1975) 77-84. | Numdam | Zbl 0368.65008

[6] M. Costabel, A coercive bilinear form for Maxwell's equations. J. Math. Anal. Appl. 157 (1991) 527-541. | Zbl 0738.35095

[7] M.L. Dudley and R.W. James, time-dependent kinematic dynamos with stationary flows. Proc. Roy. Soc. London A425 (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). | Zbl 0994.78011

[9] J.-F. Gerbeau, A stabilized finite element method for the incompressible magnetohydrodynamic equations. Numer. Math. 87 (2000) 83-111. | Zbl 0988.76050

[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). | MR 2009590 | Zbl 1037.76034

[12] V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations. Springer-Verlag, Berlin, Springer Ser. Comput. Math. 5 (1986). | Zbl 0585.65077

[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. Magnetohydrodynamics 31 (1995) 367-373. | Zbl 0875.76713

[15] J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications, Vol. 1. Dunod, Paris (1968). | MR 247243 | Zbl 0165.10801

[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. | Zbl 0948.76091

[18] J. Nečas, Les méthodes directes en théorie des équations elliptiques. Masson, Paris (1967). | MR 227584

[19] J.-C. Nédélec, A new family of mixed finite elements in ${ℝ}^{3}$. Numer. Math. 50 (1986) 57-81. | Zbl 0625.65107

[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. | Zbl 1044.76030

[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 Fluids 29 (1999) 535-554. | Zbl 0938.76049

[23] R. Verfürth, Error estimates for a mixed finite element approximation of the Stokes equation. RAIRO Anal. Numér. 18 (1984) 175-182. | Numdam | Zbl 0557.76037