, Volume 63, Issue 1, pp 315-344

Multilevel preconditioning

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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 Bramble-Pasciak-Xu preconditioner 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.

The work of this author was partially supported by the Air Force Office of Scientific Research (Contract No. 89/0455) and by the Office of Naval Research (Contract No. N00014/90/1343) during her stay at the Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA.