The use of standard nodal finite elements in the numerical solution of Maxwell's equations is marred by the occurrence of spurious modes causing severe stability problems, if no proper gauging is performed. Moreover, in case of corner or edge singularitites there might be solutions that due to a lack of regularity cannot be approximated by nodal finite elements at all. On the other hand, it is well-known that curl-conforming edge elements avoid such difficulties, since they are much closer to the variational formulation of boundary and initial-boundary value problems in electromagnetics. In this contribution, we are concerned with efficient numerical solution techniques in terms of adaptive multigrid methods based on edge element discretizations with respect to adaptively generated hierarchies of triangulations of the computational domain. In particular, we deal with multigrid algorithms whose basic ingredients are hybrid or distributive smoothing processes that take care of the nontrivial kernel of the discrete curl-operator. The characteristic feature is an additional defect correction on the subspace of irrotational vector fields. Adaptive grid refinement can be performed by means of efficient and reliable a posteriori error estimators. We present residual estimators based on an appropriate Helmholtz decomposition of the error into an irrotational and weakly solenoidal part. The performance of the multigrid algorithm and the a posteriori error estimator is illustrated by some numerical results.
