A Nonconforming Finite Element Approximation for Optimal Control of an Obstacle Problem

Asha K. Dond 1 , Thirupathi Gudi 2 ,  and Neela Nataraj 3
  • 1 Department of Mathematics, Indian Institute of Technology Bombay, Mumbai 400076, India
  • 2 Department of Mathematics, Indian Institute of Science, Bangalore 560012, India
  • 3 Department of Mathematics, Indian Institute of Technology Bombay, Mumbai 400076, India
Asha K. Dond, Thirupathi Gudi and Neela Nataraj

Abstract

The article deals with the analysis of a nonconforming finite element method for the discretization of optimization problems governed by variational inequalities. The state and adjoint variables are discretized using Crouzeix–Raviart nonconforming finite elements, and the control is discretized using a variational discretization approach. Error estimates have been established for the state and control variables. The results of numerical experiments are presented.

  • [1]

    Braess D., Finite Elements, Theory, Fast Solvers, and Applications in Solid Mechanics, Cambridge University Press, Cambridge, 1997.

  • [2]

    Brenner S. C. and Scott L. R., The Mathematical Theory of Finite Element Methods, Springer, New York, 1994.

  • [3]

    Brezzi F., Hager W. W. and Raviart P. A., Error estimates for the finite element solution of variational inequalities. Part I: Primal theory, Numer. Math. 28 (1977), 431–443.

    • Crossref
    • Export Citation
  • [4]

    Carstensen C. and Köhler K., Non-conforming FEM for the obstacle problem, IMA J Numer Anal. (2016), 10.1093/imanum/drw005.

  • [5]

    Casas E. and Tröltzsch F., Error estimates for the finite element approximation of a semi-linear elliptic control problem, Control Cybern. 31 (2002), 695–712.

  • [6]

    Falk R. S., Error estimates for the approximation of a class of variational inequalities, Math. Comp. 28 (1974), 963–971.

    • Crossref
    • Export Citation
  • [7]

    Fieldler M., Special Matrices and Their Applications in Numerical Mathematics, Martinus Nijhoff, Dordrecht, 1986.

  • [8]

    Gastaldi L. and Nochetto R., Optimal L ${L^{\infty}}$-error estimates for nonconforming and mixed finite element methods of lowest order, Numer. Math. 50 (1987), 587–611.

    • Crossref
    • Export Citation
  • [9]

    Glowinski R., Numerical Methods for Nonlinear Variational Problems, Springer, Berlin, 2008.

  • [10]

    Haslinger J. and Roubíček T., Optimal control of variational inequalities. Approximation theory and numerical realization, Appl. Math. Optim. 14 (1986), 187–201.

    • Crossref
    • Export Citation
  • [11]

    Herzog R., Rösch A., Ulbrich S. and Wollner W., OPTPDE — A collection of problems in PDE-constrained optimization, Trends in PDE Constrained Optimization, Internat. Ser. Numer. Math. 165, Springer, Cham (2014), 539–543.

  • [12]

    Hintermüller M., Ito K. and Kunisch K., The primal-dual active set strategy as a semismooth Newton method, SIAM J. Optim. 13 (2002), 865–888.

    • Crossref
    • Export Citation
  • [13]

    Hintermüller M. and Kopacka I., Mathematical programs with complementarity constraints in function space: C- and strong stationarity and a path-following algorithm, SIAM J. Optim. 20 (2009), 868–902.

    • Crossref
    • Export Citation
  • [14]

    Kinderlehrer D. and Stampacchia G., An Introduction to Variational Inequalities and Their Applications, Academic Press, New York, 1980.

  • [15]

    Kunisch K. and Wachsmuth D., Sufficient optimality conditions and semi-smooth Newton methods for optimal control of stationary variational inequalities, ESAIM Control Optim. Calc. Var. 18 (2012), 520–547.

    • Crossref
    • Export Citation
  • [16]

    Meyer C. and Thoma O., A priori finite element error analysis for optimal control of the obstacle problem, SIAM J. Numer. Anal. 51 (2013), no. 1, 605–628.

    • Crossref
    • Export Citation
  • [17]

    Mignot F. and Puel J. P., Optimal control in some variational inequalities, SIAM J. Control Optim. 22 (1984), no. 3, 466–476.

    • Crossref
    • Export Citation
  • [18]

    Nitsche J., L ${L^{\infty}}$ convergence of finite element approximations, Mathematical Aspects of Finite Element Methods, Lecture Notes in Math. 606, Springer, Berlin (1977), 261–274.

  • [19]

    Scheel H. and Scholtes S., Mathematical programs with complementarity constraints: Stationarity, optimality and sensitivity, Math. Oper. Res. 25 (2000), 1–22.

    • Crossref
    • Export Citation
  • [20]

    Wang L.-H., On the error estimate of nonconforming finite element approximation to the obstacle problem, J. Comput. Math. 21 (2003), no. 4, 481–490.

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