This paper is the second of two articles, in which we aim to prove the convergence of the Hiptmair–Xu (HX) preconditioner (originally proposed by [R. Hiptmair and J. Xu, SIAM J. Numer. Anal., 45 (2007), pp. 2483–2509]) for H(curl)-elliptic boundary value problems with jump coefficients. In this paper, based on the auxiliary results obtained in our first article [Q. Hu, SIAM J. Numer. Anal., 59 (2021), pp. 2500–2535], we establish two new regular decompositions for low-dimensional edge finite element functions in three dimensions, which, under suitable assumptions on the distribution of the coefficients, are stable up to a polylogarithmic factor of the meshwidth with respect to weighted norms defined by the coefficients. Using these regular decompositions, we analyze the convergence of the HX preconditioner for the case of strongly discontinuous coefficients. We show that the HX preconditioner is asymptotically optimal up to a polylogarithmic factor in the meshwidth and is not severely affected by large jumps of the coefficients across the interface between two neighboring subdomains. 

Publication: 

SIAM Journal on Numerical Analysis, Vol. 61, Iss. 5 (2023) 

http://dx.doi.org/10.1137/22M1501593 

Author: 

Qiya Hu 

LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China, and School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, China. 

Email: hqy@lsec.cc.ac.cn