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

Multigrid methods for 3$D$ $H(\mathbf{curl})$ problems with nonoverlapping domain decomposition smoothers