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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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Inf-sup stability implies quasi-orthogonality
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by Michael Feischl HTML | PDF
Math. Comp. 91 (2022), 2059-2094 Request permission

Abstract:

We prove new optimality results for adaptive mesh refinement algorithms for non-symmetric, indefinite, and time-dependent problems by proposing a generalization of quasi-orthogonality which follows directly from the inf-sup stability of the underlying problem. This completely removes a central technical difficulty in modern proofs of optimal convergence of adaptive mesh refinement algorithms and leads to simple optimality proofs for the Taylor-Hood discretization of the stationary Stokes problem, a finite-element/boundary-element discretization of an unbounded transmission problem, and an adaptive time-stepping scheme for parabolic equations. The main technical tools are new stability bounds for the $LU$-factorization of matrices together with a recently established connection between quasi-orthogonality and matrix factorization.
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Additional Information
  • Michael Feischl
  • Affiliation: Institute for Analysis and Scientific Computing, TU Wien, Wiedner Hauptstraße 8-10, 1040 Vienna
  • MR Author ID: 965785
  • Email: michael.feischl@tuwien.ac.at
  • Received by editor(s): July 15, 2021
  • Received by editor(s) in revised form: January 1, 2022, and March 2, 2022
  • Published electronically: July 8, 2022
  • Additional Notes: This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Project-ID 258734477 – SFB 1173 as well as the Austrian Science Fund (FWF) under the special research program Taming complexity in PDE systems (grant SFB F65)
  • © Copyright 2022 American Mathematical Society
  • Journal: Math. Comp. 91 (2022), 2059-2094
  • MSC (2020): Primary 65N30, 65N50, 15A23
  • DOI: https://doi.org/10.1090/mcom/3748
  • MathSciNet review: 4451456