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Annali di Matematica Pura ed Applicata

, Volume 117, Issue 1, pp 173–206 | Cite as

Error estimates for a galerkin approximation of a parabolic control problem

  • Ragnar Winther
Article

Summary

Numerical approximation of a parabolic control problem with a Neumann boundary condition control is considered. The observation is the final state. The numerical approximation is based on backward discretization with respect to time and a Galerkin method in the space variables. Optimal (except for a logarithmic term) L2 error estimates are derived for the optimal state. Certain error estimates for the optimal control are also given.

Keywords

Boundary Condition Optimal State Error Estimate Control Problem Numerical Approximation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    I. Babuška -A. K. Aziz,Survey lectures on the mathematical foundations of the finite element method, « The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations », A. K. Aziz, ed., Academic Press, New York, 1972, pp. 3–359.Google Scholar
  2. [2]
    G. A. Baker -J. H. Bramble -V. Thomée,Single step Galerkin approximations for parabolic problems, Math. Comp.,31, no. 140 (1977), pp. 818–847.CrossRefMathSciNetGoogle Scholar
  3. [3]
    Ju. M. Berezanskii,Expansion in eigenfunctions of self adjoint operators, Naukova Duma, Kiev, 1965; English transl., Trans. Math. Monographs, vol.17, Amer. Math. Soc., Providence, R.I., 1968.Google Scholar
  4. [4]
    J. H. Bramble -A. H. Schatz -V. Thomée -L. B. Wahlbin,Some convergence estimates for Galerkin type approximations for parabolic equations, SIAM J. Numer. Anal.14, no. 2 (1977), pp. 218–241.CrossRefMathSciNetGoogle Scholar
  5. [5]
    J. H. Bramble -V. Thomée,Discrete time Galerkin methods for a parabolic boundary value problem, Ann. Mat. Pura Appl., Serie IV,101 (1974), pp. 115–152.CrossRefMathSciNetGoogle Scholar
  6. [6]
    J. H. Bramble -V. Thomée,Semidiscrete-least squares methods for a parabolic boundary value problem, Math. Comp.,26, no. 119 (1972), pp. 633–647.CrossRefMathSciNetGoogle Scholar
  7. [7]
    J. L. Lions -E. Magenes,Non homogeneous boundary value problems and applications, vol. I–II, Springer-Verlag, New York, 1972.Google Scholar
  8. [8]
    J. L. Lions,Optimal control of systems governed by partial differential equations, Springer-Verlag, New York, 1971.Google Scholar
  9. [9]
    J. C. Nedelec,Schémas d'approximations pour des équations intégre différentielles de Riccati, Thesis, University of Paris, 1970.Google Scholar
  10. [10]
    M. Schechter,On L v estimates and regularity II, Math. Scand.,13 (1963), pp. 47–69.MATHMathSciNetGoogle Scholar
  11. [11]
    V. Thomée,Some convergence results for Galerkin methods for parabolic boundary value problems, « Mathematical Aspects of Finite Elements in Partial Differential Equations », C. de Boor ed., Academic Press, New York, 1974, pp. 55–88.Google Scholar
  12. [12]
    R. Winther,A numerical Galerkin method for a parabolic control problem, Ph. D. Thesis, Cornell University, 1977.Google Scholar

Copyright information

© Nicola Zanichelli Editore 1978

Authors and Affiliations

  • Ragnar Winther
    • 1
  1. 1.ChicagoU.S.A.

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