A Review of Numerical Analysis for the Discretization of the Velocity Tracking Problem

Chapter
Part of the SEMA SIMAI Springer Series book series (SEMA SIMAI, volume 8)

Abstract

In this paper we are reviewing results regarding the velocity tracking problem. In particular, we focus on our work (Casas and Chrysafinos, SIAM J. Numer. Anal. 50(5):2281–2306, 2012; Casas and Chrysafinos, Numer. Math. 130:615–643, 2015; and Casas and Crysafinos, to appear in ESAIM: COCV) concerning a-priori error estimates for the velocity tracking of two-dimensional evolutionary Navier-Stokes flows. The controls are of distributed type, and subject to point-wise control constraints. The standard tracking type functional is considered, however the option of setting the penalty-regularization parameter \(\lambda = 0\) in front of the L2(0, T; L2(Ω)) norm of the control in the functional is also discussed. The discretization scheme of the state and adjoint equations is based on a discontinuous time-stepping scheme combined with conforming finite elements (in space) for the velocity and pressure. Provided that the time and space discretization parameters, τ and h respectively, satisfy τ ≤ Ch2, error estimates of order \(\mathcal{O}(h)\), \(\mathcal{O}(h^{2})\) and \(\mathcal{O}(h^{\frac{3} {2} -\frac{1} {p} })\) for some p > 2, are discussed for the difference between the locally optimal controls and their discrete approximations, when the controls are discretized by piecewise constants functions, the variational discretization approach or by using piecewise-linears in space respectively for \(\lambda > 0\). For the case of \(\lambda = 0\), (bang-bang type controls) we also discuss various issues related to the analysis and discretization, emphasizing on the different features compared to the case \(\lambda > 0\). In particular, fully-discrete estimates for the states are presented and discussed.

Notes

Acknowledgements

The author “Eduardo Casas” was partially supported by the Spanish Ministerio de Economía y Competitividad under project MTM2014-57531-P.

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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Departmento de Matemática Aplicada y Ciencias de la Computación, E.T.S.I. Industriales y de TelecomunicaciónUniversidad de CantabriaSantanderSpain
  2. 2.Department of Mathematics, School of Applied Mathematics and Physical SciencesNational Technical University of AthensAthensGreece

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