Multilevel decompositions and norms for negative order Sobolev spaces

We consider multilevel decompositions of piecewise constants on simplicial meshes that are stable in H − s H^{-s} for alttext="s element-of left-parenthesis 0 comma 1 right-parenthesis"> ∈<!-- ∈ <mml:mo stretchy="false">( 0 , 1 stretchy="false">) encoding="application/x-tex">s\in (0,1) . Proofs given the case uniformly and locally refined meshes. Our findings can be applied to define local diagonal preconditioners lead bounded condition numbers (independent mesh-sizes levels) have optimal computational complexity. Furthermore, we discuss norms based (quasi-)projection operators allow efficient evaluation order Sobolev norms. Numerical examples a discussion several extensions applications conclude this article.