Error estimates for the discretization of the velocity tracking problem

Abstract

Summary In this paper we are continuing our work [6], 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 timestepping scheme (in time) 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 ≤ Ch 2 , error estimates of order O(h 2 ) and O(h 3 2 − 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.

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

“…Concerning uniqueness and error estimates under the prescribed regularity assumptions, the following results were proven in [6,Theorem 4.7], and [7,Theorem 12] Theorem 7 Given u 2 L 2 .˝T/, let us denote the solution of (2) by y 2 H 2;1 .˝T / \ C.OE0; TI Y/, and let y 2 Y be any solution of (28)- (29)- (30). Then, there exists a constant C > 0 independent of u, y and such that …”

“…In this paper we are reviewing various results from our works of [6][7][8] regarding the approximation of the velocity tracking problem. The velocity tracking problem is defined as follows: We seek velocity vector field y, pressure p and control vector field u such that Here we denote by y d the given target velocity profile and y u the solution of the 2d evolution Navier-Stokes equations with right hand side the control variable u, i.e., …”

“…Finally, in Sect. 5, we present the results of [7,8] for the discretization of the optimal control problems.…”

“…[14]) and hence the derivation of error estimates for various choices of discretization spaces for the controls when combined with piecewise constants in time, and classical finite element spaces for the velocity and the pressure (see e.g. [7]). For various related discussions, references regarding the analysis and the computational significance of such optimal control problems we refer the reader to [29].…”

“…Then, in Sect. 4, we define the discrete state and adjoint-state problems and we present the basic numerical analysis results of [6], and [7] under suitable regularity assumptions. Finally, in Sect.…”