The discretization of 3D magnetic field problems using edge-based finite elements on a tetrahedral mesh leads easily to systems with several hundreds of thousands or even millions of unknowns. The solution of such systems requires, first of all, solvers of optimal complexity. If further speed up is desired, the parallelization of the algorithms is necessary. We present parallel multigrid solvers based on a non-overlapping domain decomposition with speedups in the range of the number of processors being used. Hereby, we refer to the multigrid preconditioner as proposed by R. Hiptmair for the sequential case. However, in order to be efficient, these geometrical multigrid methods require a hierarchy of meshes. This drawback can be overcome by the application of algebraic multigrid methods (AMG). If the components, i.e., coarsening strategy, prolongation and smoother, are adapted correctly,  convergence rates which are almost  independent of the meshwidth  can be achieved by AMG also for the  curl--curl type equation discretized by Nedelec elements as being considered here. Again, a parallel version of the AMG solver can be designed based on  a non-overlapping domain decomposition. We will present performance results for both, sequential and parallel solvers. In particular,  we apply the general concept of parallelization to coupled magneto-mechanical field problems.

 * This work has been supported by the Austrian Science Fund -- 'Fonds zur Förderung der wissenschaftlichen Forschung' -- within the SFB F013  ''Numerical and Symbolic Scientific Computing''.