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

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

Research output: Journal PublicationArticlepeer-review

11 Citations (Scopus)

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.

Original languageEnglish
Pages (from-to)159-197
Number of pages39
JournalNumerische Mathematik
Volume137
Issue number1
DOIs
Publication statusPublished - 1 Sep 2017
Externally publishedYes

Keywords

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

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics

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