Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-28T03:23:51.463Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  15 August 2002

Jean Luc Guermond  and
Peter D. Minev
Jean Luc Guermond
Affiliation:
LIMSI (CNRS-UPR 3152), BP 133, 91403, Orsay, France. guermond@limsi.fr.
Peter D. Minev
Affiliation:
Department of Mathematical Sciences, University of Alberta Edmonton, Alberta, Canada T6G 2G1, Canada. minev@ualberta.ca.
Get access

Abstract

We show that the Maxwell equations in the low frequency limit, in a domain composed of insulatingand conducting regions, has a saddle point structure, wherethe electric field in the insulating region is the Lagrangemultiplier that enforces the curl-free constraint on the magnetic field.We propose a mixed finite element techniquefor solving this problem, and we show that, under mild regularityassumption on the data, Lagrange finite elements can be usedas an alternative to edge elements.

Keywords

Finite element methodMagnetohydrodynamics.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 36 , Issue 3 , May 2002 , pp. 517 - 536
Copyright
© EDP Sciences, SMAI, 2002

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Amari, T., Luciani, J.F. and Joly, P., A preconditioned semi-implicit method for magnetohydrodynamics equation. SIAM J. Sci. Comput. 21 (1999) 970-986. CrossRef
F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer Verlag, New York, Springer Ser. Comput. Math. 15 (1991).
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).
H. Brezis, Analyse fonctionnelle. Masson, Paris (1991).
Clément, P., Approximation by finite element functions using local regularization. Anal. Numér. 9 (1975) 77-84.
Costabel, M., A coercive bilinear form for Maxwell's equations. J. Math. Anal. Appl. 157 (1991) 527-541. CrossRef
Dudley, M.L. and James, R.W., time-dependent kinematic dynamos with stationary flows. Proc. Roy. Soc. London A425 (1989) 407-429. CrossRef
Demkowicz, L. and Vardapetyan, L., 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). CrossRef
Gerbeau, J.-F., A stabilized finite element method for the incompressible magnetohydrodynamic equations. Numer. Math. 87 (2000) 83-111. CrossRef
J.-L. Guermond, J. Léorat and C. Nore, Numerical simulations of 2D MHD problems using Lagrange finite elements (in preparation 2001).
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).
V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations. Springer-Verlag, Berlin, Springer Ser. Comput. Math. 5 (1986).
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.
Léorat, J., Linear dynamo simulations with time dependent helical flows. Magnetohydrodynamics 31 (1995) 367-373.
J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications, Vol. 1. Dunod, Paris (1968).
H.K. Moffatt, Magnetic Field Generation in Electrically Conducting Fluids. Cambridge Monographs on Mechanics and Applied Mathematics. Cambridge University Press, Cambridge (1978).
Meir, A.J. and Schmidt, P.G., Analysis and numerical approximation of a stationary MHD flow problem with non-ideal boundary. SIAM J. Numer. Anal. 36 (1999) 1304-1332. CrossRef
J. Necas, Les méthodes directes en théorie des équations elliptiques. Masson, Paris (1967).
Nédélec, J.-C., A new family of mixed finite elements in $\mathbb R^3$ . Numer. Math. 50 (1986) 57-81. CrossRef
Parker, R.L., Reconnexion of lines of force in rotating spheres and cylinders. Proc. Roy. Soc. 291 (1966) 60-72. CrossRef
Ben Salah, N., Soulaimani, A. and Habashi, W.G., A finite element method for magnetohydrodynamics. Comput. Methods Appl. Mech. Engrg. 190 (2001) 5867-5892. CrossRef
Ben Salah, N., Soulaimani, A., Habashi, W.G. and Fortin, M., A conservative stabilized finite element method for magnetohydrodynamics equations. Internat. J. Numer. Methods Fluids 29 (1999) 535-554. 3.0.CO;2-D>CrossRef
Verfürth, R., Error estimates for a mixed finite element approximation of the Stokes equation. RAIRO Anal. Numér. 18 (1984) 175-182. CrossRef