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Error estimates for the discretization of the velocity tracking problem

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Abstract

In this paper we are continuing our work (Casas and Chrysafinos, SIAM J Numer Anal 50(5):2281–2306, 2012), 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 discretization scheme of the state and adjoint equations is based on a discontinuous time-stepping scheme (in time) combined with conforming finite elements (in space) for the velocity and pressure. Provided that the time and space discretization parameters, \(\tau \) and \(h\) respectively, satisfy \(\tau \le Ch^2\), error estimates of order \(\mathcal {O}(h^2)\) and \(\mathcal {O}(h^{\frac{3}{2}-\frac{2}{p}})\) with \(p > 3\) depending on the regularity of the target and the initial velocity, are proved for the difference between the locally optimal controls and their discrete approximations, when the controls are discretized by the variational discretization approach and by using piecewise-linear functions in space respectively. Both results are based on new duality arguments for the evolutionary Navier–Stokes equations.

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Authors and Affiliations

  1. Departamento 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

    Eduardo Casas

  2. Department of Mathematics, School of Applied Mathematics and Physical Sciences, National Technical University of Athens, Zografou Campus, 15780, Athens, Greece

    Konstantinos Chrysafinos

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  1. Eduardo Casas
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  2. Konstantinos Chrysafinos
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Correspondence to Konstantinos Chrysafinos.

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E. Casas was partially supported by the Spanish Ministerio de Economía y Competitividad under project MTM2011-22711.

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Casas, E., Chrysafinos, K. Error estimates for the discretization of the velocity tracking problem. Numer. Math. 130, 615–643 (2015). https://doi.org/10.1007/s00211-014-0680-7

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