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

Alejandro Allendes, Enrique Otarola, Richard Rankin, Abner J. Salgado

Published 2016-08-29Version 1

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 indicator for the state equation is based on Muckenhoupt weighted Sobolev spaces, while the one for 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.