Multilevel preconditioning
type: Item
instance of: scholarly article
title: Multilevel preconditioning
zbMATH Open document ID: 0757.65031
DOI: 10.1007/BF01385864
author: Wolfgang Dahmen
author: Angela Kunoth
published in: Numerische Mathematik
publication date: 1993-02-02T00:00:00Z
full work available at URL: https://eudml.org/doc/133684
review text: This paper is concerned with multilevel techniques for preconditioning linear systems arising from Galerkin methods for elliptic boundary value problems. A general estimate is derived which is based on the characterization of Besov spaces in terms of weighted sequence norms related to corresponding multilevel expansions. The result brings out clearly how the various ingredients of a typical multilevel setting affect the growth rate of the condition numbers. In particular, our analysis indicates how to realize even uniformly bounded condition numbers. For example, the general results are used to show that the preconditioner of \textit{J. H. Bramble, J. E. Pasciak} and \textit{J. Xu} [Math. Comput. 55, No. 191, 1-22 (1990; Zbl 0703.65076)] for piecewise linear finite elements gives rise to uniformly bounded condition numbers even when the refinements of the underlying triangulations are highly nonuniform. Furthermore, they are applied to a general multivariate setting of refinable shift-invariant spaces, in particular, covering those induced by various types of wavelets.
Mathematics Subject Classification ID: 65F10
Mathematics Subject Classification ID: 65F35
Mathematics Subject Classification ID: 65N30
Mathematics Subject Classification ID: 35J25
zbMATH DE Number: 86140
zbMATH Keywords: Galerkin methods
zbMATH Keywords: condition numbers
zbMATH Keywords: multilevel representations
zbMATH Keywords: hierarchical bases
zbMATH Keywords: finite elements
zbMATH Keywords: refinable shift-invariant spaces
zbMATH Keywords: Besov spaces
MaRDI profile type: Publication
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OpenAlex ID: W2911695249
description: scientific article; zbMATH DE number 86140