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Computational Methods in Applied Mathematics

Editor-in-Chief: Carstensen, Carsten

Managing Editor: Matus, Piotr

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1609-9389
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Estimates of the Distance to the Set of Solenoidal Vector Fields and Applications to A Posteriori Error Control

Sergey Repin
• Corresponding author
• V. A. Steklov Institute of Mathematics in St. Petersburg, Fontanka 27, 191011 St. Petersburg, Russia; and University of Jyväskylä, P.O. Box 35, FI-40014, Finland
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Published Online: 2015-08-28 | DOI: https://doi.org/10.1515/cmam-2015-0024

Abstract

The paper is concerned with computable estimates of the distance between a vector-valued function in the Sobolev space ${W}^{1,\gamma }\left(\Omega ,{ℝ}^{d}\right)$ (where $\gamma \in \left(1,+\infty \right)$ and Ω is a bounded Lipschitz domain in ℝd) and the subspace ${S}^{1,\gamma }\left(\Omega ,{ℝ}^{d}\right)$ containing all divergence-free (solenoidal) vector functions. Derivation of these estimates is closely related to the stability theorem that establishes existence of a bounded operator inverse to the operator $div$. The constant in the respective stability inequality arises in the estimates of the distance to the set ${S}^{1,\gamma }\left(\Omega ,{ℝ}^{d}\right)$. In general, it is difficult to find a guaranteed and realistic upper bound of this global constant. We suggest a way to circumvent this difficulty by using weak (integral mean) solenoidality conditions and localized versions of the stability theorem. They are derived for the case where Ω is represented as a union of simple subdomains (overlapping or non-overlapping), for which estimates of the corresponding stability constants are known. These new versions of the stability theorem imply estimates of the distance to ${S}^{1,\gamma }\left(\Omega ,{ℝ}^{d}\right)$ that involve only local constants associated with subdomains. Finally, the estimates are used for deriving fully computable a posteriori estimates for problems in the theory of incompressible viscous fluids.

MSC: 65N15; 76D07; 35Q30

The author is grateful to the organizers of CMAM-6 and to the Radon Institute of Computational and Applied Mathematics in Linz.

Revised: 2015-08-16

Accepted: 2015-08-18

Published Online: 2015-08-28

Published in Print: 2015-10-01

Funding Source: RFBR

Award identifier / Grant number: N 14-01-00162

Citation Information: Computational Methods in Applied Mathematics, Volume 15, Issue 4, Pages 515–530, ISSN (Online) 1609-9389, ISSN (Print) 1609-4840,

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