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 V 1 C V 2 C... C V, we construct operators which are spectrally equivalent to those of the form A = Σ k μ k (Q k - Q k-1 ). Here μ k , k=1,2,..., are positive numbers and Q k is the orthogonal projector onto V k with Q 0 = 0. We first present abstract results which show when A is spectrally equivalent to a similarly constructed operator A defined in terms of an approximation Q k of Q k , for k = 1,2,... We show that these results lead to efficient preconditioners for discretizations of differential and pscudo-differential operators of positive and negative order. These results extend to sums of operators. For example, singularly perturbed problems such as I -eΔ can be preconditioned uniformly independently of the parameter e. 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.