Abstract: The magnetohydrodynamics (MHD) equilibrium is a fundamental concept for the plasma fusion research, allowing to describe the non-trivial steady state of a magnetic confined plasma, which is necessary to reach the physical conditions for the fusion reaction to occur. A possible mathematical formulation for finding the MHD equilibrium consists in solving a non-linear constrained optimization problem, using an energy functional as the target function. In previous works a relaxation of this task was proposed, using concepts of optimal transport and gradient flows theory. Thus, instead of solving the optimization problem directly, the minimization problem is reformulated such that an initial magnetic field is transported by a properly chosen velocity until it reaches the equilibrium. The problem can be cast into evolving the fields with the induction and the transport equation. In the continuous formulation the topology of the magnetic field is preserved along with some invariances such as helicity and a divergence-free field. In a previous work promising results with a Lagrangian scheme were shown. The goal of this work is to investigate suitable discretizations for an Eulerian scheme using geometric methods. More specifically a Spline Finite Element method is chosen and the Finite Element Exterior Calculus (FEEC) is applied. The discretizations of the transport and the induction equation are studied independently. It is shown that the discrete scheme for the transport equation with explicit time stepping is only stable for constant velocity and a Cartesian grid. For the induction equation, three different discretizations are proposed based on a strong, a weak and a mixed formulation. The discretization of the weak formulation is proven to be energy stable and the mixed formulation is proven to be energy stable and helicity preserving. Numerical results are only presented for the strong and the weak formulation, using an analytical equilibrium solution as initial condition. In comparison, the weak formulation is more stable, however both formulations can preserve the topology for a short time only.

La astronomía es la ciencia más antigua. El ser humano siempre se ha sentido fascinado por el firmamento estrellado y ha estudiado los astros con interés y dedicación, buscando siempre los recursos más adecuados. Aunque "Astronomía fundamental" no es un trabajo de divulgación, constituye una introducción accesible y muy constructiva a los grandes temas de la astrofísica y la cosmología. Presenta una descripción lúcida del contenido del universo conocido, desde los astros del Sistema Solar hasta las estrellas y las galaxias. La claridad en la exposición, así como la fluidez y la amenidad del discurso empleado, hacen que la lectura de este libro sea todo un placer para el lector.