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Computational Methods in Applied Mathematics

Editor-in-Chief: Carstensen, Carsten

Managing Editor: Matus, Piotr

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Volume 15, Issue 4


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


The paper is concerned with computable estimates of the distance between a vector-valued function in the Sobolev space W1,γ(Ω,d) (where γ(1,+) and Ω is a bounded Lipschitz domain in ℝd) and the subspace S1,γ(Ω,d) 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 S1,γ(Ω,d). 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 S1,γ(Ω,d) 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.

Keywords: inf-sup Condition; Incompressible Viscous Fluids; Domain Decomposition; A Posteriori Error Estimates

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.

About the article

Received: 2015-04-05

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, DOI: https://doi.org/10.1515/cmam-2015-0024.

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