Adaptive finite element methods for an optimal control problem involving Dirac measures

TL;DR

This paper develops and analyzes an a posteriori error estimator for an optimal control problem with Dirac measures, enabling reliable adaptive finite element methods for pointwise state tracking in 2D and 3D.

Contribution

It introduces a novel a posteriori error estimator tailored for control problems involving Dirac measures, with proven reliability and efficiency.

Findings

The error estimator performs well in numerical experiments.

Adaptive methods improve accuracy for pointwise control problems.

The analysis applies to both 2D and 3D domains.

Abstract

The purpose of this work is the design and analysis of a reliable and efficient a posteriori error estimator for the so-called pointwise tracking optimal control problem. This linear-quadratic optimal control problem entails the minimization of a cost functional that involves point evaluations of the state, thus leading to an adjoint problem with Dirac measures on the right hand side; control constraints are also considered. The proposed error estimator relies on a posteriori error estimates in the maximum norm for the state and in Muckenhoupt weighted Sobolev spaces for the adjoint state. We present an analysis that is valid for two and three-dimensional domains. We conclude by presenting several numerical experiments which reveal the competitive performance of adaptive methods based on the devised error estimator.