Abstract: This paper is intended as a survey of current results on algorithmic and theoretical aspects of overlapping Schwarz methods for discrete $\Hcurl$ and $\Hdiv$--elliptic problems set in suitable finite element spaces. The emphasis is on a unified framework for the motivation and theoretical study of the various approaches developed in recent years.

Generalized Helmholtz decompositions -- orthogonal decompositions into the null space of the relevant differential operator and its complement -- are crucial in our considerations. It turns out that the decompositions the Schwarz methods are based upon have to be designed separately for both components. In the case of the null space, the construction has to rely on liftings into spaces of discrete potentials.

Taking the cue from well-known Schwarz schemes for second order elliptic problems, we devise uniformly stable splittings of both parts of the Helmholtz decomposition. They immediately give rise to powerful preconditioners and iterative solvers.