Stabilized Lagrangian finite elements in electromagnetism and in incompressible magnetohydrodynamics

This thesis deals with the usage of nodal-based finite element methods in electromagnetism and incompressible magnetohydrodynamics. At first, we start to investigate the induction equation from Maxwell's equation. We give a stability and semidiscrete error analysis in space for this problem. Additionally, a further force term from a predescribed velocity field is added such that we end up with an extended induction equation. We also perform a stability and semidiscrete error analysis in space for this model which is stabilized with a so-called local projection stabilization for the additional force term and stabilization for the divergence free constraint of the magnetic field. Finally, the Navier-Stokes equations are coupled with the extended induction equation and we get the model of resistive incompressible magnetohydrodynamics. This problem is stationarized and linearized. A stability and semidiscrete error analysis is performed in this case. The stabilization is needed for the mass conservation of the velocity field, the divergence free constraint of the magnetic field, the magnetic pseudo-pressure which is introduced as a Lagrangian multiplier and the nonlinear terms appearing in the Navier-Stokes equations and the extended induction equation. We distinguish the cases of equal-order and inf-sup-stable finite element pairs in this thesis. Finally, some numerical examples underline the performance of our proposed method