Variational-FEEC discretization for the equations of MagnetoHydroDynamics

We propose a new class of finite element approximations to the compressible magnetohydrodynamics equations. Our discretizations are built via a discrete least action principle mimicking the continuous Euler-Poincaré principle, allowing the preservation of important structures of the problem. We will also shortly describe the inclusion of dissipative terms in this framework, first on their inclusion in the continuous least principle and then on the discrete one. The resulting semi-discrete approximations are shown to conserve the total mass, and energy of the solutions and create entropy in the presence of dissipative terms. In addition the divergence-free nature of the magnetic field is preserved in a pointwise sense. Numerical simulations are conducted, using spline finite elements (IGA), to verify the accuracy of our approach, its ability to preserve the semi-discrete invariants for several test problem and then test its performance on the simulation of plasma instability in simplified tokamak geometry