Computational scales of Sobolev norms with application to preconditioning

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 à 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− ∆ 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.