A new practical framework for the stability analysis of perturbed saddle-point problems and applications
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- by Qingguo Hong, Johannes Kraus, Maria Lymbery and Fadi Philo HTML | PDF
- Math. Comp. 92 (2023), 607-634 Request permission
Abstract:
In this paper we prove a new abstract stability result for perturbed saddle-point problems based on a norm fitting technique. We derive the stability condition according to Babuška’s theory from a small inf-sup condition, similar to the famous Ladyzhenskaya-Babuška-Brezzi (LBB) condition, and the other standard assumptions in Brezzi’s theory, in a combined abstract norm. The construction suggests to form the latter from individual fitted norms that are composed from proper seminorms.
This abstract framework not only allows for simpler (shorter) proofs of many stability results but also guides the design of parameter-robust norm-equivalent preconditioners. These benefits are demonstrated on mixed variational formulations of generalized Poisson, Stokes, vector Laplace and Biot’s equations.
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Additional Information
- Qingguo Hong
- Affiliation: Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania 16802
- MR Author ID: 1007990
- Email: huq11@psu.edu
- Johannes Kraus
- Affiliation: Faculty of Mathematics, University of Duisburg-Essen, Thea-Leymann-Straße 9, Essen 45127, Germany
- MR Author ID: 672969
- ORCID: 0000-0003-2261-599X
- Email: johannes.kraus@uni-due.de
- Maria Lymbery
- Affiliation: Faculty of Mathematics, University of Duisburg-Essen, Thea-Leymann-Straße 9, Essen 45127, Germany
- Email: maria.lymbery@uni-due.de
- Fadi Philo
- Affiliation: Faculty of Mathematics, University of Duisburg-Essen, Thea-Leymann-Straße 9, Essen 45127, Germany
- MR Author ID: 1330732
- Email: fadi.philo@uni-due.de
- Received by editor(s): May 27, 2021
- Received by editor(s) in revised form: April 27, 2022, and August 28, 2022
- Published electronically: November 17, 2022
- Additional Notes: The second author and the third author were supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) as part of the project “Physics-oriented solvers for multicompartmental poromechanics” under grant number 456235063
- © Copyright 2022 American Mathematical Society
- Journal: Math. Comp. 92 (2023), 607-634
- MSC (2020): Primary 65N12, 65J05, 65F08, 65N30
- DOI: https://doi.org/10.1090/mcom/3795
- MathSciNet review: 4524104