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Petrov-Galerkin Crank-Nicolson Scheme for Parabolic Optimal Control Problems on Nonsmooth Domains

  • Thomas G. Flaig
  • Dominik MeidnerEmail author
  • Boris Vexler
Chapter
Part of the International Series of Numerical Mathematics book series (ISNM, volume 165)

Abstract

In this paper we transfer the a priori error analysis for the discretization of parabolic optimal control problems on domains allowing for H 2 regularity (i.e. either with smooth boundary or polygonal and convex) to a large class of nonsmooth domains. We show that a combination of two ingredients for the optimal convergence rates with respect to the spatial and the temporal discretization is required. First we need a time discretization scheme which has the desired convergence rate in the smooth case. Secondly we need a method to treat the singularities due to non-smoothness of the domain for the corresponding elliptic state equation. In particular we demonstrate this philosophy with a Crank-Nicolson time discretization and finite elements on suitably graded meshes for the spatial discretization. A numerical example illustrates the predicted convergence rates.

Keywords

Optimal control problem Parabolic partial differential equation Non-smooth domains Graded meshes Crank Nicolson scheme 

Mathematics Subject Classification (2010)

49M25 49M05 65M15 65M60 49M29 65M12 

Notes

Acknowledgements

The function u s for the numerical example in Sect. 6 was taken from a presentation by T. Apel and J. Pfefferer. The numerical experiments in Sect. 6 are carried out using both a MATLAB-FEM implementation based on [1, 12, 15] and the software packages RODOBO [30] and GASCOIGNE [16].

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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Thomas G. Flaig
    • 1
  • Dominik Meidner
    • 2
    Email author
  • Boris Vexler
    • 2
  1. 1.Institut für Mathematik und BauinformatikUniversität der Bundeswehr MünchenNeubibergGermany
  2. 2.Fakultät für Mathematik, Lehrstuhl für Optimale SteuerungTechnische Universität MünchenGarching b. MünchenGermany

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