The analysis of multigrid methods (Q2702612)

The paper is one main part in the ``Handbook of Numerical Analysis'' and devoted to the analysis of multigrid methods. While the first chapter is devoted to an introduction to iterative methods and the two-level multigrid method, Chapter II presents the abstract theory of multigrid methods. The V-cycle multigrid method is analyzed in detail and multilevel preconditioners are looked at. The analysis of classes of smoothers finishes this chapter.NEWLINENEWLINENEWLINEThe largest chapter, Chapter III with nearly 100 pages is devoted to applications. From second-order elliptic problems to pseudodifferential operators the reader finds detailed mathematical analysis concerning the multigrid method. It is a pitty that only 10 pages are concerned with practical issues of actual implementations but the overall size of the book most likely has ruled out any deatiled study necessary for practioneers. Chapter IV contains two appendices concerned with interpolation spaces and a glossary of conditions containing a condensed description of the theorems outlined in the text.NEWLINENEWLINENEWLINEAs the preface states this paper is an expanded and updated version of \textit{J. H. Bramble's} book [Multigrid methods (1993; Zbl 0786.65094)] and based on a graduate course. In the preface the reader also can find a description of this paper which matches perfectly my own impressions: ``It is meant to introduce the reader to theoretical considerations involved in the study of multigrid methods keeping the practical applicability of the abstract theory in mind. The results are presented in a uniform abstract setting''.NEWLINENEWLINEFor the entire collection see [Zbl 0953.00016].