Localized forms of the LBB condition and a posteriori estimates for incompressible media problems
Repin, S. (2018). Localized forms of the LBB condition and a posteriori estimates for incompressible media problems. Mathematics and Computers in Simulation, 145, 156-170. doi:10.1016/j.matcom.2016.05.004
Published inMathematics and Computers in Simulation
© 2016 International Association for Mathematics and Computers in Simulation (IMACS). This is a final draft version of an article whose final and definitive form has been published by Elsevier. Published in this repository with the kind permission of the publisher.
The inf–sup (or LBB) condition plays a crucial role in analysis of viscous flow problems and other problems related to incompressible media. In this paper, we deduce localized forms of this condition that contain a collection of local constants associated with subdomains instead of one global constant for the whole domain. Localized forms of the LBB inequality imply estimates of the distance to the set of divergence free fields. We use them and deduce fully computable bounds of the distance between approximate and exact solutions of boundary value problems arising in the theory of viscous incompressible fluids. The estimates are valid for approximations, which satisfy the incompressibility condition only in a very weak (integral) form. Another important question considered in the paper is how to select proper measures that should be used in error analysis. We show that such a measure is dictated by the respective error identity and discuss properties of the measure for the Stokes, Oseen, and Navier–Stokes problems. ...