Computational scales of Sobolev norms with application to preconditioning

This paper provides a framework for developing computation- ally ecient multilevel preconditioners and representations for Sobolev norms. Specically, given a Hilbert space V and a nested sequence of subspaces V1 V2 ::: V , we construct operators which are spectrally equiva- lent to those of the formA = P kk(Qk Qk 1). Here k, k =1 ; 2;::: ,a re positive numbers andQk is the orthogonal projector onto Vk with Q0 =0 . We rst present abstract results which show whenA is spectrally equivalent to a similarly constructed operator e A dened in terms of an approximation e Qk of Qk ,f ork =1 ; 2;::: . We show that these results lead to ecient preconditioners for discretiza- tions of dierential and pseudo-dierential operators of positive and negative order. These results extend to sums of operators. For example, singularly per- turbed problems such as I can be preconditioned uniformly independently of the parameter . We also show how to precondition an operator which re- sults from Tikhonov regularization of a problem with noisy data. Finally, we describe how the technique provides computationally ecient bounded discrete extensions which have applications to domain decomposition.