An a posteriori error analysis for an optimal control problem with point sources

TL;DR

This paper develops a reliable a posteriori error estimator for a linear-quadratic optimal control problem with point sources, enabling effective adaptive methods in 2D and 3D.

Contribution

It introduces a novel a posteriori error estimator combining weighted Sobolev spaces and maximum norm techniques for problems with Dirac measures.

Findings

The estimator is reliable and efficient for 2D and 3D domains.

Adaptive strategy based on the estimator achieves optimal convergence rates.

Numerical examples confirm the effectiveness of the proposed approach.

Abstract

We propose and analyze a reliable and efficient a posteriori error estimator for a constrained linear-quadratic optimal control problem involving Dirac measures; the control variable corresponds to the amplitude of forces modeled as point sources. The proposed a posteriori error estimator is defined as the sum of two contributions, which come from the state and adjoint equations. The estimator associated with the state equation is based on Muckenhoupt weighted Sobolev spaces, while the one associated with the adjoint is in the maximum norm and allows for unbounded right hand sides. The analysis is valid for two and three-dimensional domains. On the basis of the devised a posteriori error estimator, we design a simple adaptive strategy that yields optimal rates of convergence for the numerical examples that we perform.