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

Convergence of HX Preconditioner for Maxwell's Equations with Jump Coefficients (i): Various Extensions of The Regular Helmholtz Decomposition