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

TL;DR

This paper extends the Helmholtz decomposition for edge finite element functions in 3D domains, facilitating the proof of HX preconditioner convergence for Maxwell's equations with jump coefficients.

Contribution

It develops new Helmholtz decomposition extensions that preserve boundary conditions and stability estimates, crucial for analyzing preconditioner convergence.

Findings

Helmholtz decompositions preserve zero tangential complement on domain boundaries.

Decompositions possess stability estimates with only a logarithmic factor.

Foundation for proving HX preconditioner convergence in subsequent work.

Abstract

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).