Construction principles of multigrid smoothers for Curl-Curl equations

The construction principle for multigrid smoothers for discrete Curl-Curl equations consists in the inclusion of an additional zero divergence constraint. This principle is shown to hold for established schemes such as Hiptmair's hybrid smoother and it is used to construct a new smoother starting from a mixed formulation using a Lagrange multiplier formulation, where a zero divergence constraint leads to a grad-div augmented system. The convergence properties of this system are compared to the nonaugmented system and to the hybrid scheme using Gauss-Seidel iterations for different curl-curl systems arising from various electrodynamical problems.