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This paper is the second one of two serial articles, whose goal is to proveconvergence of HX Preconditioner (proposed by Hiptmair and Xu, 2007) forMaxwell's equations with jump coefficients. In this paper, based on theauxiliary results developed in the first paper (Hu, 2017), we establish a newregular Helmholtz decomposition for edge finite element functions in threedimensions, which is nearly stable with respect to a weight function. By usingthis Helmholtz decomposition, we give an analysis of the convergence of the HXpreconditioner for the case with strongly discontinuous coefficients. We showthat the HX preconditioner possesses fast convergence, which not only is nearlyoptimal with respect to the finite element mesh size but also is independent ofthe jumps in the coefficients across the interface between two neighboringsubdomains.

Convergence of HX Preconditioner for Maxwell's Equations with Jump Coefficients (ii): The Main Results