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 V and a nested sequence of subspaces V 1 ⊂ V 2 ⊂ … ⊂ V V_1 ⊂ V_2 ⊂ … ⊂ V , we construct operators which are spectrally equivalent to those of the form A = ∑ k μ k ( Q k − Q k − 1 ) A= ∑ _k μ _k (Q_k-Qₖ₋₁) . Here μ k μ _k , k = 1 , 2 , … k=1,2,… , are positive numbers and Q k Q_k is the orthogonal projector onto V k V_k with Q 0 = 0 Q_0=0 . We first present abstract results which show when A A is spectrally equivalent to a similarly constructed operator A ~