Open AccessMultilevel decompositions and norms for negative order Sobolev spaces
Author(s) -
Thomas Führer
Publication year - 2021
Publication title -
mathematics of computation
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.95
H-Index - 103
eISSN - 1088-6842
pISSN - 0025-5718
DOI - 10.1090/mcom/3674
Subject(s) - algorithm , computer science , sobolev space , mathematics , artificial intelligence , mathematical analysis
We consider multilevel decompositions of piecewise constants on simplicial meshes that are stable in H − s H^{-s} for s ∈ ( 0 , 1 ) s\in (0,1) . Proofs are given in the case of uniformly and locally refined meshes. Our findings can be applied to define local multilevel diagonal preconditioners that lead to bounded condition numbers (independent of the mesh-sizes and levels) and have optimal computational complexity. Furthermore, we discuss multilevel norms based on local (quasi-)projection operators that allow the efficient evaluation of negative order Sobolev norms. Numerical examples and a discussion on several extensions and applications conclude this article.