A discrete weighted Helmholtz decomposition and its application

We propose a discrete weighted Helmholtz decomposition for edge element functions. The decomposition is orthogonal in a weighted \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document} inner product and stable uniformly with respect to the jumps in the discontinuous weight function. As an application, the new Helmholtz decomposition is applied to demonstrate the quasi-optimality of a preconditioned edge element system for solving a saddle-point Maxwell system in non-homogeneous media by a non-overlapping domain decomposition preconditioner, i.e., the condition number grows only as the logarithm of the dimension of the local subproblem associated with an individual subdomain, and more importantly, it is independent of the jumps of the physical coefficients across the interfaces between any two subdomains of different media. Numerical experiments are presented to validate the effectiveness of the non-overlapping domain decomposition preconditioner.