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Adaptive edge element approximation of H(curl)-elliptic optimal control problems with control constraints

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Abstract

A three-dimensional H(curl)-elliptic optimal control problem with distributed control and pointwise constraints on the control is considered. We present a residual-type a posteriori error analysis with respect to a curl-conforming edge element approximation of the optimal control problem. Here, the lowest order edge elements of Nédélec’s first family are used for the discretization of the state and the control with respect to an adaptively generated family of simplicial triangulations of the computational domain. In particular, the a posteriori error estimator consists of element and face residuals associated with the state equation and the adjoint state equation. The main results are the reliability of the estimator and its efficiency up to oscillations in terms of the data of the problem. In the last part of the paper, numerical results are included which illustrate the performance of the adaptive approach.

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Acknowledgments

The first author acknowledges support by the NSF Grants DMS-1115658, DMS-1216857, by the German National Science Foundation within the Priority Programs SPP 1253, SPP 1506, by the German Federal Ministry for Education and Research (BMBF) within the projects BMBF-FROPT and BMBF-MeFreSim, and by the European Science Foundation (ESF) within the ESF Program OPTPDE.

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Authors and Affiliations

  1. Department of Mathematics, University of Houston, Houston, TX, 77204-3008, USA

    Ronald H. W. Hoppe

  2. Institute of Mathematics, University of Augsburg, 86159 , Augsburg, Germany

    Ronald H. W. Hoppe

  3. Graduate School of Excellence Computational Engineering, Technical University Darmstadt, 64293 , Darmstadt, Germany

    Irwin Yousept

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  1. Ronald H. W. Hoppe
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  2. Irwin Yousept
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Correspondence to Ronald H. W. Hoppe.

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Communicated by Ulrich Langer.

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Hoppe, R.H.W., Yousept, I. Adaptive edge element approximation of H(curl)-elliptic optimal control problems with control constraints. Bit Numer Math 55, 255–277 (2015). https://doi.org/10.1007/s10543-014-0497-x

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  • DOI: https://doi.org/10.1007/s10543-014-0497-x

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