Journal of Scientific Computing

, Volume 60, Issue 1, pp 160–183 | Cite as

Adaptive Finite Element Approximation for an Elliptic Optimal Control Problem with Both Pointwise and Integral Control Constraints



In this paper, we study the adaptive finite element approximation for a constrained optimal control problem with both pointwise and integral control constraints. We first obtain the explicit solutions for the variational inequalities both in the continuous and discrete cases. Then a priori error estimates are established, and furthermore equivalent a posteriori error estimators are derived for both the state and the control approximation, which can be used to guide the mesh refinement for an adaptive multi-mesh finite element scheme. The a posteriori error estimators are implemented and tested with promising numerical results.


Optimal control problem Finite element approximation  Adaptive finite element method A posteriori error estimates Multi-meshes 


  1. 1.
    Ainsworth, M., Oden, J.T.: A posteriori error estimators in finite element analysis. Comput. Methods Appl. Mech. Eng. 142, 1–88 (1997)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Becker, R., Kapp, H., Rannacher, R.: Adaptive finite element methods for optimal control of partial differential equations: basic concept. SIAM J. Control Optim. 39, 113–132 (2000)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Becker, R., Rannacher, R.: An optimal control approach to a-posteriori error estimation. In: Iserles, A. (ed.) Acta Numerica 2001, pp. 1–102. Cambridge University Press, Cambridge (2001)Google Scholar
  4. 4.
    Bejan, A.: Constructal-theory network of conducting paths for cooling a heat generating volume. Int. J. Heat Mass Transf. 40(4), 779–816 (1997)CrossRefGoogle Scholar
  5. 5.
    Casas, E., Troltzsch, F.: Second order necessary and sufficient optimality conditions for optimization problems and applications to control theory. SIAM J. Optim. 12, 406–431 (2002)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)MATHGoogle Scholar
  7. 7.
    Deckelnick, K., Hinze, M.: Variational discretization of parabolic control problems in the presence of pointwise state constraints. J. Comp. Math. 29, 1–15 (2011)MATHMathSciNetGoogle Scholar
  8. 8.
    Dorfler, W.: A convergent adaptive algorithm for Poisson’s equations. SIAM J. Numer. Anal. 33, 1106–1124 (1996)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Ge, L., Liu, W.B., Yang, D.P.: Adaptive finite element approximation for a constrained optimal control problem via multi-meshes. J. Sci. Comput. 41, 238–255 (2009)Google Scholar
  10. 10.
    Guo, Z.Y., Cheng, X.G., Xia, Z.Z.: Least dissipation principle of heat transport potential capacity and its application in heat conduction optimization. Chin. Sci. Bull. 48(4), 406–410 (2003)CrossRefGoogle Scholar
  11. 11.
    Heinkenschloss, K., Vicente, L.N.: Analysis fo inexact trust-region SQP algorithms. SIAM J. Optim. 12, 283–302 (2001)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Kelley, C.T., Sachs, E.W.: A trust region method for parabolic boundary control problems. SIAM J. Optim. 9, 1064–1091 (1999)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Kufner, A., John, O., Fucik, S.: Function Spaces. Nordhoff, Leiden (1977)MATHGoogle Scholar
  14. 14.
    Lee, H., Lee, J.: A stochastic Galerkin method for stochastic control problems. Commun. Comput. Phys. 14, 77–106 (2013)MathSciNetGoogle Scholar
  15. 15.
    Li, R., Liu, W.B., Ma, H.P., Tang, T.: Adaptive finite element approximation of elliptic optimal control. SIAM J. Control. Optim. 41, 1321–1349 (2002)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Li, R.: On Multi-Mesh h-Adaptive Algorithm. J. Sci. Comput. 24, 321–341 (2005)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Lions, J.L.: Optimal Control of Systems Governed by Partial Differential Equations. Springer, Berlin (1971)CrossRefMATHGoogle Scholar
  18. 18.
    Liu, W.B., Tiba, D.: Error estimates for the finite element approximation of a class of nonlinear optimal control problems. Numer. Funct. Anal. Optim. 22, 953–972 (2001)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Liu, W.B., Yan, N.N.: A posteriori error analysis for convex distributed optimal control problems. Adv. Comput. Math. 15(1–4), 285–309 (2001)CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Liu, W.B., Yan, N.N.: Adaptive Finite Element Methods for Optimal Control Governed by PDEs, p. 1. Science Press, Beijing (2008)Google Scholar
  21. 21.
    Liu, W.B., Yang, D.P., Yuan, L., Ma, C.Q.: Finite element approximations of an optimal control problem with integral state constraint. SIAM J. Numer. Anal. 48, 1163–1185 (2010)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Morin, P., Nochetto, R.H., Siebert, K.G.: Data oscillation and convergence of adaptive FEM. SIAM J. Numer. Anal. 38, 466–488 (2001)CrossRefMathSciNetGoogle Scholar
  23. 23.
    Niu, H., Yang, D.: Finite element analysis of optimal control problem governed by Stokes equations with \(L^2\)-norm state-constraints. J. Comput. Math. 29, 589–604 (2011)CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    Roos, H., Reibiger, C.: Numerical analysis of a system of singularly perturbed convection-diffusion equations related to optimal control. Numer. Math. Theor. Meth. Appl. 4, 562–575 (2011)MATHMathSciNetGoogle Scholar
  25. 25.
    Scott, L.R., Zhang, S.: Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput. 54, 483–493 (1990)CrossRefMATHMathSciNetGoogle Scholar
  26. 26.
    Vallejos, M.: Multigrid methods for elliptic optimal control problems with pointwise state Constraints. Numer. Math. Theor. Meth. Appl. 5, 99–109 (2012)MATHMathSciNetGoogle Scholar
  27. 27.
    Veeser, A.: Efficient and reliable a posteriori error estimators for elliptic obstacle problems. SIAM J. Numer. Anal. 39(1), 146–167 (2001)CrossRefMATHMathSciNetGoogle Scholar
  28. 28.
    Verfürth, R.: A posteriori error estimators for the Stokes equations. Numer. Math. 55, 309–325 (1989)CrossRefMATHMathSciNetGoogle Scholar
  29. 29.
    Verfürth, R.: A Review of A Posteriori Error Estimation and Adaptive Mesh Refinement. Wiley-Teubner, London (1996)MATHGoogle Scholar

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© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.School of MathematicsShandong UniversityJinanChina
  2. 2.Institute of Thermal Science and TechnologyShandong UniversityJinanChina
  3. 3.KBSUniversity of KentCanterburyUK

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