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

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Abstract

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.

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References

  1. Ainsworth, M., Oden, J.T.: A posteriori error estimators in finite element analysis. Comput. Methods Appl. Mech. Eng. 142, 1–88 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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. Bejan, A.: Constructal-theory network of conducting paths for cooling a heat generating volume. Int. J. Heat Mass Transf. 40(4), 779–816 (1997)

    Article  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  6. Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)

    MATH  Google Scholar 

  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)

    MATH  MathSciNet  Google Scholar 

  8. Dorfler, W.: A convergent adaptive algorithm for Poisson’s equations. SIAM J. Numer. Anal. 33, 1106–1124 (1996)

    Article  MathSciNet  Google Scholar 

  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. 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)

    Article  Google Scholar 

  11. Heinkenschloss, K., Vicente, L.N.: Analysis fo inexact trust-region SQP algorithms. SIAM J. Optim. 12, 283–302 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  12. Kelley, C.T., Sachs, E.W.: A trust region method for parabolic boundary control problems. SIAM J. Optim. 9, 1064–1091 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  13. Kufner, A., John, O., Fucik, S.: Function Spaces. Nordhoff, Leiden (1977)

    MATH  Google Scholar 

  14. Lee, H., Lee, J.: A stochastic Galerkin method for stochastic control problems. Commun. Comput. Phys. 14, 77–106 (2013)

    MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  16. Li, R.: On Multi-Mesh h-Adaptive Algorithm. J. Sci. Comput. 24, 321–341 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  17. Lions, J.L.: Optimal Control of Systems Governed by Partial Differential Equations. Springer, Berlin (1971)

    Book  MATH  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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. 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)

    Article  MATH  MathSciNet  Google Scholar 

  22. Morin, P., Nochetto, R.H., Siebert, K.G.: Data oscillation and convergence of adaptive FEM. SIAM J. Numer. Anal. 38, 466–488 (2001)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    MATH  MathSciNet  Google Scholar 

  25. Scott, L.R., Zhang, S.: Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput. 54, 483–493 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  26. Vallejos, M.: Multigrid methods for elliptic optimal control problems with pointwise state Constraints. Numer. Math. Theor. Meth. Appl. 5, 99–109 (2012)

    MATH  MathSciNet  Google Scholar 

  27. Veeser, A.: Efficient and reliable a posteriori error estimators for elliptic obstacle problems. SIAM J. Numer. Anal. 39(1), 146–167 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  28. Verfürth, R.: A posteriori error estimators for the Stokes equations. Numer. Math. 55, 309–325 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  29. Verfürth, R.: A Review of A Posteriori Error Estimation and Adaptive Mesh Refinement. Wiley-Teubner, London (1996)

    MATH  Google Scholar 

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Authors and Affiliations

  1. School of Mathematics, Shandong University, Jinan, Shandong, China

    Ning Du

  2. Institute of Thermal Science and Technology, Shandong University, Jinan, Shandong, China

    Liang Ge

  3. KBS, University of Kent, Canterbury, CT2 7NF, UK

    Wenbin Liu

Authors
  1. Ning Du
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  2. Liang Ge
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  3. Wenbin Liu
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Corresponding author

Correspondence to Wenbin Liu.

Additional information

Ning Du was supported by the NSFC under the Grant 11371229 and 11201265.

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Du, N., Ge, L. & Liu, W. Adaptive Finite Element Approximation for an Elliptic Optimal Control Problem with Both Pointwise and Integral Control Constraints. J Sci Comput 60, 160–183 (2014). https://doi.org/10.1007/s10915-013-9790-0

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  • DOI: https://doi.org/10.1007/s10915-013-9790-0

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