PhD Seminars XI

We are dealing with the numerical approximation of magnetohydrodynamic (MHD) instabilities of an incompressible ionized gas (plasma) in the tokamaks (a toroidal device in which plasma is confined thanks to a very strong applied magnetic field). In the approximation used by physicists, these models involve two divergence-free vectors : the magnetic field and velocity. By writing these vector fields as rotationals of two stream functions, we obtain a system of partial differential equations in which differential terms of order 4 appear. The conforming approximation of these terms requires finite elements with C1 regularity. The use of stream functions provides a natural way to achieve the divergence-free constraints. We also assume that the flows are independent of the toroidal coordinate and the resulting model is thus 2-D. We have implemented a C1 finite element method on general triangles : the so-called Clough Tocher finite element. The developed method is applied to solve the tilting MHD instabilities and plasma equilibrium in tokamaks in axisymmetric configuration. Results are shown to agree with other studies and theoretical scaling results.