An Adaptive FEM for the Pointwise Tracking Optimal Control Problem of the Stokes Equations

Abstract

We propose and analyze a reliable and efficient a posteriori error estimator for the pointwise tracking optimal control problem of the Stokes equations. This linear-quadratic optimal control problem entails the minimization of a cost functional that involves point evaluations of the velocity field that solves the state equations. This leads to an adjoint problem with a linear combination of Dirac measures as a forcing term and whose solution exhibits reduced regularity properties. We also consider constraints on the control variable. The proposed a posteriori error estimator can be decomposed as the sum of four contributions: three contributions related to the discretization of the state and adjoint equations, and another contribution that accounts for the discretization of the control variable. On the basis of the devised a posteriori error estimator, we design a simple adaptive strategy that illustrates our theory and exhibits a competitive performance. L 2 (ω,G) 1 2 .

Finite Element Error Estimation for Parabolic Optimal Control Problems with Pointwise Observations

“…By using standard arguments (cf. [29]) we can prove that the optimization problem (3.1) admits a unique solution ū ∈ U ad for any β ∈ [1,2]. The first order necessary (also sufficient) optimality condition of the optimization problem (3.1) at ū reads as follows:…”

“…Although there are already some works on the error estimates of elliptic or Stokes control problems with pointwise tracking (cf. [2,4,5,7,9,15]), to the authors' best knowledge, there are no such results for the parabolic control problems with pointwise tracking. For the discretization of the state and adjoint equations, we use a piecewise constant discontinuous Galerkin scheme (the DG(0) method) for the temporal discretization, and the standard linear finite element method (the CG(1) method) for the spatial discretization.…”

“…, N 2 } are respectively called the spatial observations and the time observations. The parameter β ∈ [1,2] is the weight between the pointwise spatial observations and pointwise time observations. Specifically, β = 1 refers to the case of pure spatial observation and β = 2 refers to the case of pure time observation, while β ∈ (1, 2) refers to the case with both observations and weights the importance of two observations.…”

“…The optimization problem (1.3) with cost functional of pointwise type serves as a model problem for several applications, such as the parameter identification or inverse problems with finitely many pointwise measurements (cf. [35]), and optimal control problems where the target states on some spatial or time points are of particular interest or can be measured by sensors [2,4,9]. We also refer the reader to [29] for the derivation of the necessary optimality conditions, and to [2,4,5,7,9,15,35,39] for the numerical analysis of this kind of problems.…”

“…[35]), and optimal control problems where the target states on some spatial or time points are of particular interest or can be measured by sensors [2,4,9]. We also refer the reader to [29] for the derivation of the necessary optimality conditions, and to [2,4,5,7,9,15,35,39] for the numerical analysis of this kind of problems.…”