The numerical simulation of transient magneto-quasistatic fields ranges among the most challenging problems from the point of modelling and numerical linear algebra in Computational Electromagnetics. Using the Finite Integration Technique a magneto-quasistatic formulation is considered, in which a curlcurl-formulation with a modified magnetic vector potential results in a degenerated parabolic system of equations. The degenerate character arises from the large nullspace of the curlcurl-stiffness matrix and the typical singularity of the material matrix of conductivities. The regularization of the formulation, the so-called "gauging", involves techniques to control the irrotational parts of the vector potential in the non-conductive regions. For the resulting systems of differential-algebraic equations of Index 1 implicit time integration methods such as Gear's BDF-methods or stiffly accurate SDIRK-methods are applicable. Stiff integrators with embedded schemes, such as e.g. the L-stable, 3rd order Rosenbrock-Wanner-3(2) method, which also provides an embedded scheme of 2nd order, allow to use sophisticated error estimation schemes for variable time stepping. 

Ferromagnetic material behavior in eddy current simulations requires a nonlinear algebraic system to be solved at each time step. Linearization methods such as the Successive-Step algorithm with underrelaxation or a Newton-Raphson-scheme typically increase the arithmetical work per time step significantly.
The SSOR-preconditioned Conjugate Gradient method provides a robust solver for the repetitive solution of the consistently singular linear systems, which is denoted as "weak gauging property". However, since the asymptotical complexity of this method is not optimal, large problems with millions of unknowns will require extensive calculation times.
Multigrid methods, which have to be especially adapted to the singular curlcurl equations, show the required asymptotical optimal complexity property and therefor is suited also for large numbers of degrees of freedom.

Eddy current effects introduced by motion of conducting materials are to be included, if  e.g. linear motors or eddy current brake systems are considered. Suitably augmented eddy current formulations depend on the choice of the coordinate system for the moving conductor: The Eulerian coordinate description, where the coordinate system is fixed to the field excitation source, results in an additional advective term in the system matrix. Here generalized CG-type methods for non-Hermitean linear systems are to be applied. The stability of the scheme depends on the motion velocities to be suitably slow.
In Lagrangian coordinate description, where the coordinate system is fixed to the moving conductor, only the right hand sides of the time discretized systems are modified and the system matrices remain symmetric.