Geodesic
Magnetic equations with FreeFem++: The Grad-Shafranov equation the current hole
Erwan Deriaz1 ; Bruno Despres2 ; Gloria Faccanoni3 ; Kirill Pichon Gostaf2 ; Lise-Marie Imbert-Gérard2 ; Georges Sadaka4 ; Remy Sart5
1 Laboratoire de Mécanique, Modélisation et Procédés Propres, 38 rue Frédéric Joliot-Curie 13451 Marseille, France
2 Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, 75252 Paris, France
3 Institut de mathématiques de Toulon et du Var, Université du Sud Toulon-Var, 83957 La Garde, France
4 Laboratoire Amiénois de Mathématique Fondamentale et Appliquée, Université de Picardie Jules Verne, 33 rue Saint-Leu, 80039 Amiens, France
5 École Supérieure d’Ingénieurs Léonard de Vinci, 92916 Paris, France
ESAIM. Proceedings, Tome 32 (2011), pp. 76-94.

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FreeFem++ [11] is a software for the numerical solution of partial differential equations. It is based on finite element method. The FreeFem++ platform aims at facilitating teaching and basic research through prototyping. For the moment this platform is restricted to the numerical simulations of problems which admit a variational formulation. Our goal in this work is to evaluate the FreeFem++ tool on basic magnetic equations arising in Fusion Plasma in the context of the ITER project. First we consider the Grad-Shafranov equation, which is derived from the static ideal MHD equations assuming axisymetry. Some of the properties of the equation and its analytical solutions are discussed. Second we discretize a reduced resistive MHD model which admits solutions of the Grad-Shafranov equation as stationary solutions. Then the physical stability of these stationary solutions is investigated through numerical experiments and the numerical stability of the algorithm is discussed.
DOI : 10.1051/proc/2011013

Erwan Deriaz 1 ; Bruno Despres 2 ; Gloria Faccanoni 3 ; Kirill Pichon Gostaf 2 ; Lise-Marie Imbert-Gérard 2 ; Georges Sadaka 4 ; Remy Sart 5

1 Laboratoire de Mécanique, Modélisation et Procédés Propres, 38 rue Frédéric Joliot-Curie 13451 Marseille, France
2 Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, 75252 Paris, France
3 Institut de mathématiques de Toulon et du Var, Université du Sud Toulon-Var, 83957 La Garde, France
4 Laboratoire Amiénois de Mathématique Fondamentale et Appliquée, Université de Picardie Jules Verne, 33 rue Saint-Leu, 80039 Amiens, France
5 École Supérieure d’Ingénieurs Léonard de Vinci, 92916 Paris, France 
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     title = {Magnetic equations with {FreeFem++:} {The} {Grad-Shafranov} equation & the current hole},
     journal = {ESAIM. Proceedings},
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Erwan Deriaz; Bruno Despres; Gloria Faccanoni; Kirill Pichon Gostaf; Lise-Marie Imbert-Gérard; Georges Sadaka; Remy Sart. Magnetic equations with FreeFem++: The Grad-Shafranov equation & the current hole. ESAIM. Proceedings, Tome 32 (2011), pp. 76-94. doi : 10.1051/proc/2011013. http://geodesic.mathdoc.fr/articles/10.1051/proc/2011013/

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