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

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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.

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Acknowledgements

The authors would like to thank Harbir Antil (George Mason University) for insightful discussions.

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Correspondence to Enrique Otárola.

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AA is partially supported by USM through Project 116.12.1. EO is supported by CONICYT through FONDECYT Project 3160201 and Anillo ACT-1106. RR is supported by BASAL PFB03 CMM project, Universidad de Chile. AJS is partially supported by NSF Grant DMS-1418784. Part of this work was carried out while EO was visiting the University of Tennessee through a special visitors program for the academic year 2015–2016.

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Allendes, A., Otárola, E., Rankin, R. et al. Adaptive finite element methods for an optimal control problem involving Dirac measures. Numer. Math. 137, 159–197 (2017). https://doi.org/10.1007/s00211-017-0867-9

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