  This paper presents a numerical study on multigrid algorithms of $V$-cycle
type for problems posed in the Hilbert space $H(\mathbf{curl})$ in three
dimensions. The multigrid methods are designed for discrete problems originated
from the discretization using the hexahedral N\'{e}d\'{e}lec edge element of
the lowest-order. Our suggested methods are associated with smoothers
constructed by substructuring based on domain decomposition methods of
nonoverlapping type. Numerical experiments to demonstrate the robustness and
the effectiveness of the suggested algorithms are also provided.
