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A priori \(L^2\)-discretization error estimates for the state in elliptic optimization problems with pointwise inequality state constraints

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Abstract

In this paper, an elliptic optimization problem with pointwise inequality constraints on the state is considered. The main contributions of this paper are a priori \(L^2\)-error estimates for the discretization error in the optimal states. Due to the non separability of the space for the Lagrange multipliers for the inequality constraints, the problem is tackled by separation of the discretization error into two components. First, the state constraints are discretized. Second, with discretized inequality constraints, a duality argument for the error due to the discretization of the PDE is employed. For the second stage an a priori error estimate is derived with constants depending on the regularity of the dual problem. Finally, we discuss two cases in which these constants can be bounded in a favorable way; leading to higher order estimates than those induced by the known \(L^2\)-error in the control variable. More precisely, we consider a given fixed number of pointwise inequality constraints and a case of infinitely many but only weakly active constraints.

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Acknowledgements

The authors would like to thank Gerd Wachsmuth for pointing out an issue in a prior version of this manuscript. Moreover I. Neitzel and W. Wollner are grateful for the support of their former host institutions the Technische Universität München and the Universität Hamburg, respectively.

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Authors and Affiliations

  1. Institut für Numerische Simulation, Rheinische Friedrich-Wilhelms-Universität Bonn, Wegelerstr. 6, 53115, Bonn, Germany

    I. Neitzel

  2. Fachbereich Mathematik, Technische Universität Darmstadt, Dolivostr. 15, 64293, Darmstadt, Germany

    W. Wollner

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  1. I. Neitzel
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  2. W. Wollner
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Correspondence to I. Neitzel.

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Neitzel, I., Wollner, W. A priori \(L^2\)-discretization error estimates for the state in elliptic optimization problems with pointwise inequality state constraints. Numer. Math. 138, 273–299 (2018). https://doi.org/10.1007/s00211-017-0906-6

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  • DOI: https://doi.org/10.1007/s00211-017-0906-6

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