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Numerische Mathematik

, Volume 137, Issue 1, pp 159–197 | Cite as

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

  • Alejandro Allendes
  • Enrique OtárolaEmail author
  • Richard Rankin
  • Abner J. Salgado
Article

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.

Keywords

Pointwise tracking optimal control problem Dirac measures A posteriori error analysis Adaptive finite elements Maximum norm Muckenhoupt weights Weighted Sobolev spaces 

Mathematics Subject Classification

49J20 49M25 65K10 65N15 65N30 65N50 65Y20 

Notes

Acknowledgements

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

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Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  • Alejandro Allendes
    • 1
  • Enrique Otárola
    • 1
    Email author
  • Richard Rankin
    • 1
  • Abner J. Salgado
    • 2
  1. 1.Departamento de MatemáticaUniversidad Técnica Federico Santa MaríaValparaísoChile
  2. 2.Department of MathematicsUniversity of TennesseeKnoxvilleUSA

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