Model problems in magneto-hydrodynamics: Individual numerical challenges and coupling possibilities

In this work we discuss two model problems appearing in magneto-hydrodynamics (MHD), namely, the so called full MHD problem and the inductionless MHD problem. The first involves as unknowns the fluid velocity and pressure, the magnetic (induction) field and a pseudo-pressure introduced to impose the zero-divergence restriction of this last unknown. The building blocks of this model are the Stokes problem for the velocity and the pressure and the Maxwell problem for the magnetic field and pseudo-pressure. We discuss the numerical challenges of the approximation of these two model problems having in mind the need to couple them in the full problem, where additional coupling terms appear. The second model we consider is the inductionless MHD approximation. Instead of the magnetic induction and pseudo-pressure, the magnetic unknowns are now the current density and the electric potential. The building blocks are the Stokes problem for the fluid and the Darcy problem (in primal form) for the current density and electric potential. We discuss also the numerical challenges involved in the approximation of this last problem, particularly considering that it has to be coupled to the former. Once the building blocks have been analysed independently, the possibilities of dealing with the fully coupled problems are discussed. Iterative schemes that can be shown to be stable are presented in the stationary case, showing that a segregated solution for the flow and the magnetic problem is not possible. Most of the results presented are elaborated independently in previous works. Our objective in this paper is to present the different problems with a unified perspective.