Multilevel Preconditioning

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 aaect 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 nite elements gives rise to uniformly bounded condition numbers even when the reenements of the underlying triangulations are highly nonuniform. Furthermore , they are applied to a general multivariate setting of reenable shift-invariant spaces, in particular, covering those induced by various types of wavelets.

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