Algebraic Theory of Two-Grid Methods. Issue 2 (28th May 2015)

• Record Type: Journal Article
• Authors: Notay, Yvan
• Is Part Of: Numerical mathematics. Volume 8:Issue 2(2015)
• Page Start: 168
• Page End: 198
• Publication Date: 2015-05-28
• Subjects: Numerical analysis -- Periodicals; Numerical analysis; Periodicals; 518.05
• DOI: 10.4208/nmtma.2015.w04si
• Languages: English
• ISSNs: 1004-8979

Abstract

<abstract abstract-type="normal"> <title>Abstract</title> <p>About thirty years ago, Achi Brandt wrote a seminal paper providing a convergence theory for algebraic multigrid methods [Appl. Math. Comput., 19 (1986), pp. 23–56]. Since then, this theory has been improved and extended in a number of ways, and these results have been used in many works to analyze algebraic multigrid methods and guide their developments. This paper makes a concise exposition of the state of the art. Results for symmetric and nonsymmetric matrices are presented in a unified way, highlighting the influence of the smoothing scheme on the convergence estimates. Attention is also paid to sharp eigenvalue bounds for the case where one uses a single smoothing step, allowing straightforward application to deflation-based preconditioners and two-level domain decomposition methods. Some new results are introduced whenever needed to complete the picture, and the material is self-contained thanks to a collection of new proofs, often shorter than the original ones.</p> </abstract>