Volume 52, Number 5, September–October 2018
|Page(s)||1617 - 1650|
|Published online||22 November 2018|
An a posteriori error analysis for an optimal control problem with point sources★
Departamento de Matemática, Universidad Técnica Federico Santa María,
2 School of Mathematical Sciences, University of Nottingham Ningbo China, Ningbo, China.
3 Department of Mathematics, University of Tennessee, Knoxville, TN 37996, USA.
* Corresponding author: email@example.com
Accepted: 22 January 2018
We propose and analyze a reliable and efficient a posteriori error estimator for a control-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 are associated with 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.
Mathematics Subject Classification: 49J20 / 49M25 / 65K10 / 65N15 / 65N30 / 65Y20
Key words: Linear-quadratic optimal control problem / Dirac measures / a posteriori error analysis / adaptive finite elements / maximum norm / Muckenhoupt weights / weighted Sobolev spaces
© EDP Sciences, SMAI 2018
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