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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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Multilevel decompositions and norms for negative order Sobolev spaces
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by Thomas Führer HTML | PDF
Math. Comp. 91 (2022), 183-218 Request permission

Abstract:

We consider multilevel decompositions of piecewise constants on simplicial meshes that are stable in $H^{-s}$ for $s\in (0,1)$. 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.
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Additional Information
  • Thomas Führer
  • Affiliation: Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Santiago, Chile
  • MR Author ID: 1017746
  • ORCID: 0000-0001-5034-6593
  • Email: tofuhrer@mat.uc.cl
  • Received by editor(s): August 31, 2020
  • Received by editor(s) in revised form: April 22, 2021
  • Published electronically: August 12, 2021
  • Additional Notes: This work was supported by ANID through FONDECYT projects 11170050 and 1210391.
  • © Copyright 2021 American Mathematical Society
  • Journal: Math. Comp. 91 (2022), 183-218
  • MSC (2020): Primary 65F08, 65F35, 65N30, 65N38
  • DOI: https://doi.org/10.1090/mcom/3674
  • MathSciNet review: 4350537