In this paper, a multigrid method is introduced to effectively solve the linear system resulting from a finite element discretization for problems posed in \(H(\textbf{curl})\) . The hexahedral edge element of the lowest order forms the basis for the discretization. Due to the inapplicability of classical multigrid methods for vector field problems, smoothers of the Schwarz type, which typically requires high computational expenses, have been considered. The primary objective is to propose a smoother that is cost-efficient. The convergence analysis for the multigrid method we suggest is presented. In addition, we provide numerical examples to validate our theory and to demonstrate the effectiveness of the method.
