This paper is the second one of two serial articles, whose goal is to prove convergence of HX Preconditioner (proposed by Hiptmair and Xu, 2007) for Maxwell's equations with jump coefficients. In this paper, based on the auxiliary results developed in the first paper (Hu, 2017), we establish a new regular Helmholtz decomposition for edge finite element functions in three dimensions, which is nearly stable with respect to a weight function. By using this Helmholtz decomposition, we give an analysis of the convergence of the HX preconditioner for the case with strongly discontinuous coefficients. We show that the HX preconditioner possesses fast convergence, which not only is nearly optimal with respect to the finite element mesh size but also is independent of the jumps in the coefficients across the interface between two neighboring subdomains.

翻译：本文是两篇系列文章中的第二篇,这两篇系列文章的目标是证明HX预设物(由Hiptmair和Xu提出,2007年)与Maxwell的公式与跳系数的趋同。在本文中,我们根据第一份论文(Hu, 2017年)中得出的辅助结果,建立了一个新的定期Helmholtz分解法,用于三个维度的边缘有限元素功能,这在重量函数方面几乎稳定。通过使用Helmholtz分解法,我们分析了本案HX预设物与高度不连续系数的趋同情况。我们表明,HX预设物具有快速趋同性,这不仅与有限元素中位大小几乎是最佳的,而且独立于两个相邻子域界面的系数跳动。