Computational scales of Sobolev norms with application to preconditioning
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This paper provides a framework for developing computationally efficient multilevel preconditioners and representations for Sobolev norms. Specifically, given a Hilbert space V and a nested sequence of subspaces V1 ⊂ V2 ⊂ . . . ⊂ V, we construct operators which are spectrally equivalent to those of the form A = ∑k μk (Qk - Qk-1). Here μk, k = 1, 2, . . . , are positive numbers and Qk is the orthogonal projector onto Vk with Q0 = 0. We first present abstract results which show when A is spectrally equivalent to a similarly constructed operator Ã defined in terms of an approximation Q̃k of Qk, for k = 1, 2, . . . . We show that these results lead to efficient preconditioners for discretizations of differential and pseudo-differential operators of positive and negative order. These results extend to sums of operators. For example, singularly perturbed problems such as I - ∈Δ can be preconditioned uniformly independently of the parameter ∈. We also show how to precondition an operator which results from Tikhonov regularization of a problem with noisy data. Finally, we describe how the technique provides computationally efficient bounded discrete extensions which have applications to domain decomposition.
author list (cited authors)
Bramble, J. H., Pasciak, J. E., & Vassilevski, P. S.