Error estimates for the approximation of the velocity tracking problem with Bang-Bang controls^∗

¹ Departmento de Matemática Aplicada y Ciencias de la Computación, E.T.S.I. Industriales y de Telecomunicación, Universidad de Cantabria, Av. Los Castros s/n, 39005 Santander, Spain.
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² Department of Mathematics, School of Applied Mathematics and Physical Sciences, National Technical University of Athens, Zografou Campus, 15780 Athens, Greece.
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Received: 29 May 2015
Revised: 22 December 2015
Accepted: 2 August 2016

Abstract

The velocity tracking problem for the evolutionary Navier–Stokes equations in 2d is studied. The controls are of distributed type but the cost functional does not involve the usual quadratic term for the control. As a consequence the resulting controls can be of bang-bang type. First and second order necessary and sufficient conditions are proved. A fully-discrete scheme based on discontinuous (in time) Galerkin approach combined with conforming finite element subspaces in space, is proposed and analyzed. Provided that the time and space discretization parameters, τ and h respectively, satisfy τ ≤ Ch², then L² error estimates are proved for the difference between the states corresponding to locally optimal controls and their discrete approximations.