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，旨在证明HX预条件子（由Hiptmair和Xu于2007年提出）对含跳跃系数麦克斯韦方程的收敛性。基于第一篇文章（Hu, 2017）中发展的辅助结果，我们建立了三维情况下边有限元函数的一种新的正则亥姆霍兹分解，该分解关于权函数近似稳定。利用这一亥姆霍兹分解，我们分析了强间断系数情形下HX预条件子的收敛性。研究表明，HX预条件子具有快速收敛性，不仅相对于有限元网格尺寸近似最优，而且与相邻子域界面处系数的跳跃无关。