Mimetic spectral element method for the Grad-Shafranov equation A. Palha 1 , F. Felici 1 , B. Koren 1 1 Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands Motivation The computation of magnetohydrodynamic (MHD) equilibrium of toroidal plasmas plays a central role in tokamak fusion studies, not only as a fundamental tool for the computation of transport and stability calculations but also for the development of advanced tokamak control systems. Under perfect axisymmetric conditions, the vector calculus formulation of the Grad- Shafranov equation is the one most commonly used, [1, 2]: −∇ ·  1 μ r ∇ψ  = r d p dψ + 1 μ 0 r f d f dψ := j t . (1) This equation can be written as a system of three equations:      ∇ψ =  u ∇ ·  h = ˜ j t and K  u =  h with K :=   1 μ 0 r 0 0 1 μ 0 r   . (2) The two equations on the left define differential (topological) relations and the third one es- tablishes a constitutive relation between  u (the gradient of ψ ) and the flux vector  h. The direct discretization of this formulation using standard techniques presents two challenges: • Conservation of current at the discrete level (  Ω ∇ ·  h dΩ =  Ω j t dΩ). • Extension to curved meshes. The first one is an important requirement for coupling with transport equations, the second enables optimal high order approximation in curved domains. Differential Geometry Formulation In this work, we propose to use the differential geometric formulation of this equation: d ⋆ K d ⋆ ˜ ψ (2) = ˜ j (2) t . (3) This corresponds to two topological relations (left) and two constitutive relations (right):      dψ (0) = u (1) d ˜ h (1) = ˜ j (2) t and      ⋆ ˜ ψ (2) = ψ (0) K ⋆ u (1) = ˜ h (1) , (4) 42 nd EPS Conference on Plasma Physics P1.185