Variational Multi-Scale stabilized Finite Elements for the magnetohydrodynamic models of fusion plasmas

The main objective of this thesis concerns the implementation of a robust stabilized finite element method for simulating fusion plasmas. For that, we first derive the magnetohydrodynamic models from the kinetic model. MHD models are generally used for macroscopic simulations of plasma instabilities. We concentrate ou efforts on the full MHD model. Next, the numerical approach is described in the context of the Variational Multi-Scale (VMS) stabilization. This stabilization comes to add a term to the weak formulation to mimic the effects of the unresolved scales over the coarse scales. If the effects of these sub-scales are not taken into account when dealing with fluxes dominated by convection, as it is the cases for fusion plasmas, the numerical scheme can lead to unphysical results. A detailed study of the resistive internal kink instability has been done as well as an introductory study of the so called X-point plasmas in order to validate the numerical scheme developed here. (author)