Two-grid methods with exact solution of the Galerkin coarse-grid system have been well studied by the multigrid community: an elegant identity has been established to characterize the convergence factor of exact two-grid methods. In practice, however, it is often too costly to solve the Galerkin coarse-grid system exactly, especially when its size is large. Instead, without essential loss of convergence speed, one may solve the coarse-grid system approximately. In this paper, we develop a new framework for analyzing the convergence of inexact two-grid methods: two-sided bounds for the energy norm of the error propagation matrix of inexact two-grid methods are presented. In the framework, a restricted smoother involved in the identity for exact two-grid convergence is used to measure how far the actual coarse-grid matrix deviates from the Galerkin one. As an application, we establish a new and unified convergence theory for multigrid methods. 

　　Publication: 

　　SIAM Journal on Matrix Analysis and Applications, Volume 43, Issue 1 

　　https://doi.org/10.1137/21M140448X 

　　Author: 

　　Xuefeng Xu 

　　Department of Mathematics, Purdue University, West Lafayette, IN47907, USA 

　　Chensong Zhang 

　　LSEC & NCMIS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China 

　　School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China 

　　Email: zhangcs@lsec.cc.ac.cn