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

TL;DR

This paper proves the convergence of the HX preconditioner for Maxwell's equations with jump coefficients, demonstrating its efficiency and robustness even with highly discontinuous material properties.

Contribution

It introduces a new Helmholtz decomposition for edge finite element functions, enabling the analysis of the HX preconditioner's convergence with discontinuous coefficients.

Findings

HX preconditioner converges rapidly for Maxwell's equations with jump coefficients

Convergence is nearly optimal and independent of coefficient jumps

Helmholtz decomposition is nearly stable with respect to a weight function

Abstract

This paper is the second 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, based on the auxiliary results developed in the first paper (Hu, 2017), we establish a new regular Helmholtz decomposition for edge finite element functions in three dimensions, which is nearly stable with respect to a weight function. By using this Helmholtz decomposition, we give an analysis of the convergence of the HX preconditioner for the case with strongly discontinuous coefficients. We show that the HX preconditioner possesses fast convergence, which not only is nearly optimal with respect to the finite element mesh size but also is independent of the jumps in the coefficients across the interface between two neighboring subdomains.