Numerical Analysis
Mixed finite elements for incompressible magneto-hydrodynamics
[Méthode d'éléments finis mixtes pour la magnéto-hydrodynamique incompressible]
Comptes Rendus. Mathématique, Tome 337 (2003) no. 1, pp. 71-74.

Nous présentons une nouvelle méthode d'éléments finis mixtes pour les équations stationnaires tridimensionnelles de la magnéto-hydrodynamique incompressible. La partie fluide est discrétisée par des couples d'espaces standards vitesse–pression, stables selon la condition inf–sup, et la partie magnétique par une approche mixte utilisant les éléments de Nédélec de première espèce. Nous montrons que la méthode qui en résulte converge de façon quasi-optimale.

We present a new mixed finite element discretization for three-dimensional stationary incompressible magneto-hydrodynamics. The fluid variables are discretized by standard inf–sup stable velocity–pressure pairs and the magnetic variables by a mixed approach using Nédélec's elements of the first kind. The resulting method is shown to be quasi-optimally convergent.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/S1631-073X(03)00256-5
Schneebeli, Anna 1 ; Schötzau, Dominik 1

1 Department of Mathematics, University of Basel, Rheinsprung 21, CH-4051 Switzerland
@article{CRMATH_2003__337_1_71_0,
     author = {Schneebeli, Anna and Sch\"otzau, Dominik},
     title = {Mixed finite elements for incompressible magneto-hydrodynamics},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {71--74},
     publisher = {Elsevier},
     volume = {337},
     number = {1},
     year = {2003},
     doi = {10.1016/S1631-073X(03)00256-5},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1016/S1631-073X(03)00256-5/}
}
TY  - JOUR
AU  - Schneebeli, Anna
AU  - Schötzau, Dominik
TI  - Mixed finite elements for incompressible magneto-hydrodynamics
JO  - Comptes Rendus. Mathématique
PY  - 2003
SP  - 71
EP  - 74
VL  - 337
IS  - 1
PB  - Elsevier
UR  - http://archive.numdam.org/articles/10.1016/S1631-073X(03)00256-5/
DO  - 10.1016/S1631-073X(03)00256-5
LA  - en
ID  - CRMATH_2003__337_1_71_0
ER  - 
%0 Journal Article
%A Schneebeli, Anna
%A Schötzau, Dominik
%T Mixed finite elements for incompressible magneto-hydrodynamics
%J Comptes Rendus. Mathématique
%D 2003
%P 71-74
%V 337
%N 1
%I Elsevier
%U http://archive.numdam.org/articles/10.1016/S1631-073X(03)00256-5/
%R 10.1016/S1631-073X(03)00256-5
%G en
%F CRMATH_2003__337_1_71_0
Schneebeli, Anna; Schötzau, Dominik. Mixed finite elements for incompressible magneto-hydrodynamics. Comptes Rendus. Mathématique, Tome 337 (2003) no. 1, pp. 71-74. doi : 10.1016/S1631-073X(03)00256-5. http://archive.numdam.org/articles/10.1016/S1631-073X(03)00256-5/

[1] Armero, F.; Simo, J.C. Long-term dissipativity of time-stepping algorithms for an abstract evolution equation with applications to the incompressible MHD and Navier–Stokes equations, Comput. Methods Appl. Mech. Engrg., Volume 131 (1996), pp. 41-90

[2] Costabel, M.; Dauge, M. Weighted regularization of Maxwell equations in polyhedral domains, Numer. Math., Volume 93 (2002), pp. 239-277

[3] Demkowicz, L.; Vardapetyan, L. Modeling of electromagnetic absorption/scattering problems using hp-adaptive finite elements, Comput. Methods Appl. Mech. Engrg., Volume 152 (1998), pp. 103-124

[4] Gerbeau, J.-F. A stabilized finite element method for the incompressible magnetohydrodynamic equations, Numer. Math., Volume 87 (2000), pp. 83-111

[5] Guermond, J.-L.; Minev, P. Mixed finite element approximation of an MHD problem involving conducting and insulating regions: the 2D case, Math. Model. Numer. Anal., Volume 36 (2002), pp. 517-536

[6] J.-L. Guermond, P. Minev, Mixed finite element approximation of an MHD problem involving conducting and insulating regions: the 3D case, Numer. Methods Partial Differential Equations, to appear

[7] Gunzburger, M.D.; Meir, A.J.; Peterson, J.S. On the existence and uniqueness and finite element approximation of solutions of the equations of stationary incompressible magnetohydrodynamics, Math. Comp., Volume 56 (1991), pp. 523-563

[8] Nédélec, J.C. Mixed finite elements in 3 , Numer. Math., Volume 35 (1980), pp. 315-341

[9] D. Schötzau, Mixed finite element methods for stationary incompressible magneto-hydrodynamics, Tech. report 2003-03, Department of Mathematics, University of Basel

[10] R. Hartmann, W. Bangerth, G. Kanschat, deal.II differential equations analysis library, technical reference, IWR, Universität Heidelberg, http://www.dealii.org

Cité par Sources :