Error estimates for the discretization of the velocity tracking problem

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 $$h$$h respectively, satisfy $$\tau \le Ch^2$$?≤Ch2, error estimates of order $$\mathcal {O}(h^2)$$O(h2) and $$\mathcal {O}(h^{\frac{3}{2}-\frac{2}{p}})$$O(h32-2p) with $$p > 3$$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.