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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.
Allendes, Alejandro 1 ; Otárola, Enrique 1 ; Rankin, Richard 1 ; Salgado, Abner J. 1
@article{M2AN_2018__52_5_1617_0,
author = {Allendes, Alejandro and Ot\'arola, Enrique and Rankin, Richard and Salgado, Abner J.},
title = {An a posteriori error analysis for an optimal control problem with point sources},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1617--1650},
publisher = {EDP-Sciences},
volume = {52},
number = {5},
year = {2018},
doi = {10.1051/m2an/2018010},
zbl = {1415.49002},
mrnumber = {3878607},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/m2an/2018010/}
}
TY - JOUR AU - Allendes, Alejandro AU - Otárola, Enrique AU - Rankin, Richard AU - Salgado, Abner J. TI - An a posteriori error analysis for an optimal control problem with point sources JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2018 SP - 1617 EP - 1650 VL - 52 IS - 5 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/m2an/2018010/ DO - 10.1051/m2an/2018010 LA - en ID - M2AN_2018__52_5_1617_0 ER -
%0 Journal Article %A Allendes, Alejandro %A Otárola, Enrique %A Rankin, Richard %A Salgado, Abner J. %T An a posteriori error analysis for an optimal control problem with point sources %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2018 %P 1617-1650 %V 52 %N 5 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/m2an/2018010/ %R 10.1051/m2an/2018010 %G en %F M2AN_2018__52_5_1617_0
Allendes, Alejandro; Otárola, Enrique; Rankin, Richard; Salgado, Abner J. An a posteriori error analysis for an optimal control problem with point sources. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 5, pp. 1617-1650. doi: 10.1051/m2an/2018010
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