  This paper is the first 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 we establish various
extensions of the regular Helmholtz decomposition for edge finite element
functions defined in three dimensional domains. The functions defined by the
regular Helmholtz decompositions can preserve the zero tangential complement on
faces and edges of polyhedral domains and some non-Lipchitz domains, and
possess stability estimates with only a $logarithm$ factor. These regular
Helmholtz decompositions will be used to prove convergence of the HX
preconditioner for Maxwell's equations with jump coefficients in another paper
(Hu 2017).
