Adaptive discontinuous Galerkin methods for state constrained optimal control problems governed by convection diffusion equations

Abstract

We study a posteriori error estimates for the numerical approximations of state constrained optimal control problems governed by convection diffusion equations, regularized by Moreau–Yosida and Lavrentiev-based techniques. The upwind Symmetric Interior Penalty Galerkin (SIPG) method is used as a discontinuous Galerkin (DG) discretization method. We derive different residual-based error indicators for each regularization technique due to the regularity issues. An adaptive mesh refinement indicated by a posteriori error estimates is applied. Numerical examples are presented to illustrate the effectiveness of the adaptivity for both regularization techniques.

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Acknowledgments

The authors would like to thank Martin Stoll for helpful discussions about this research. This work was supported by the Forschungszentrum Dynamische Systeme (CDS): Biosystemtechnik, Otto-von-Guericke-Universität Magdeburg. The authors also would like to express their sincere thanks to the referees for most valuable suggestions.

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Correspondence to Hamdullah Yücel.

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Yücel, H., Benner, P. Adaptive discontinuous Galerkin methods for state constrained optimal control problems governed by convection diffusion equations. Comput Optim Appl 62, 291–321 (2015). https://doi.org/10.1007/s10589-014-9691-7

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Keywords

  • Optimal control problem
  • State constraints
  • Discontinuous Galerkin methods
  • Convection diffusion equations
  • A posteriori error estimates