Adaptive finite element methods for the pointwise tracking optimal control problem of the Stokes equations

TL;DR

This paper develops and analyzes an a posteriori error estimator for adaptive finite element methods addressing a pointwise tracking optimal control problem governed by Stokes equations, accounting for reduced regularity due to Dirac measures.

Contribution

It introduces a reliable, efficient a posteriori error estimator for a complex control problem with pointwise velocity tracking and control constraints, enabling adaptive refinement.

Findings

The estimator effectively guides adaptive mesh refinement.

The adaptive method shows competitive performance in numerical tests.

The approach handles reduced regularity from Dirac measures successfully.

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.