  We propose V--cycle multigrid methods for vector field problems arising from
the lowest order hexahedral N\'{e}d\'{e}lec finite element. Since the
conventional scalar smoothing techniques do not work well for the problems, a
new type of smoothing method is necessary. We introduce new smoothers based on
substructuring with nonoverlapping domain decomposition methods. We provide the
convergence analysis and numerical experiments that support our theory.
