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.

翻译：我们建议采用V-周期多格方法来解决由六等分母最小的六等分母N\'{{e}d\{e}lec有限元素产生的矢量场问题。由于传统的平滑天平技术不能很好地解决问题,因此有必要采用一种新的平滑方法。我们引入基于非重叠域分解方法的亚结构的新的平滑器。我们提供了支持我们理论的趋同分析和数字实验。