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 CONVERGENCE ANALYSIS OF INEXACT TWO-GRID METHODS: A THEORETICAL FRAMEWORK Xu, Xuefeng1; Zhang, Chen-Song2,3,4 2022 Source Publication SIAM JOURNAL ON NUMERICAL ANALYSIS ISSN 0036-1429 Volume 60Issue:1Pages:133-156 Abstract Multigrid is one of the most efficient methods for solving large-scale linear systems that arise from discretized partial differential equations. As a foundation for multigrid analysis, two-grid theory plays an important role in motivating and analyzing multigrid algorithms. For symmetric positive definite problems, the convergence theory of two-grid methods with exact solution of the Galerkin coarse-grid system is mature, and the convergence factor of exact two-grid methods can be characterized by an identity. Compared with the exact case, the convergence theory of inexact two-grid methods (i.e., the coarse-grid system is solved approximately) is of more practical significance, while it is still less developed in the literature (one reason is that the error propagation matrix of inexact coarse-grid correction is not a projection). In this paper, we develop a theoretical framework for the convergence analysis of inexact two-grid methods. More specifically, we present two-sided bounds for the energy norm of the error propagation matrix of inexact two-grid methods, from which one can readily obtain the identity for exact two-grid convergence. As an application, we establish a unified convergence theory for multigrid methods, which allows the coarsest-grid system to be solved approximately. Keyword multigrid inexact two-grid methods convergence factor eigenvalue analysis DOI 10.1137/20M1356075 Indexed By SCI Language 英语 Funding Project National Key R\&D Program of China[2020YFA0711900] ; National Key R\&D Program of China[2020YFA0711904] ; National Science Foundation of China[11971472] ; Key Research Program of Frontier Sciences of CAS WOS Research Area Mathematics WOS Subject Mathematics, Applied WOS ID WOS:000765864500006 Publisher SIAM PUBLICATIONS Citation statistics Document Type 期刊论文 Identifier http://ir.amss.ac.cn/handle/2S8OKBNM/60185 Collection 计算数学与科学工程计算研究所 Corresponding Author Xu, Xuefeng Affiliation 1.Purdue Univ, Dept Math, W Lafayette, IN 47907 USA2.Chinese Acad Sci, Acad Math & Syst Sci, LSEC, Beijing 100190, Peoples R China3.Chinese Acad Sci, Acad Math & Syst Sci, NCMIS, Beijing 100190, Peoples R China4.Univ Chinese Acad Sci, Sch Math Sci, Beijing 100049, Peoples R China Recommended CitationGB/T 7714 Xu, Xuefeng,Zhang, Chen-Song. CONVERGENCE ANALYSIS OF INEXACT TWO-GRID METHODS: A THEORETICAL FRAMEWORK[J]. SIAM JOURNAL ON NUMERICAL ANALYSIS,2022,60(1):133-156. APA Xu, Xuefeng,&Zhang, Chen-Song.(2022).CONVERGENCE ANALYSIS OF INEXACT TWO-GRID METHODS: A THEORETICAL FRAMEWORK.SIAM JOURNAL ON NUMERICAL ANALYSIS,60(1),133-156. MLA Xu, Xuefeng,et al."CONVERGENCE ANALYSIS OF INEXACT TWO-GRID METHODS: A THEORETICAL FRAMEWORK".SIAM JOURNAL ON NUMERICAL ANALYSIS 60.1(2022):133-156.
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