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Convergence and quasi-optimality of an adaptive finite element method for optimal control problems with integral control constraint

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Abstract

In this paper we study the convergence of an adaptive finite element method for optimal control problems with integral control constraint. For discretization, we use piecewise constant discretization for the control and continuous piecewise linear discretization for the state and the co-state. The contraction, between two consecutive loops, is proved. Additionally, we find the adaptive finite element method has the optimal convergence rate. In the end, we give some examples to support our theoretical analysis.

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References

  1. Babuška, I., Rheinboldt, W.C.: Error estimates for adaptive finite element computations. SIAM J. Numer. Anal. 15, 736–754 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  2. Dörfler, W.: A convergent adaptive algorithm for Poisson’s equation. SIAM J. Numer. Anal. 33, 1106–1124 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  3. Morin, P., Nochetto, R.H., Siebert, K.G.: Data oscillation and convergence of adaptive FEM. SIAM J. Numer. Anal. 38, 466–488 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  4. Morin, P., Nochetto, R.H., Siebert, K.G.: Convergence of adaptive finite element methods. SIAM Rev. 44, 631–658 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chen, Z., Feng, J.: An adaptive finite element algorithm with reliable and efficient error control for linear parabolic problems. Math. Comp. 73, 1167–1193 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  6. Mekchay, K., Nochetto, R.H.: Convergence of adaptive finite element methods for general second order linear elliptic PDEs. SIAM J. Numer. Anal. 43, 1803–1827 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  7. Morin, P., Siebert, K.G., Veeser, A.: A basic convergence result for conforming adaptive finite elements. Math. Mod. Meth. Appl. S. 18, 707–737 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cascon, J.M., Kreuzer, C., Nochetto, R.H., Siebert, K.G.: Qusi-optimal convergence rate for an adaptive finite element method. SIAM J. Numer. Anal. 46, 2524–2550 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  9. Binev, P., Dahmen, W., Devore, R.: Adaptive finite element methods with convergence rates. Numer. Math. 97, 219–268 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  10. Stevenson, R.: Optimality of a standard adaptive finite element method. Found Comput. Math. 7, 245–269 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  11. Carstensen, C., Hoppe, R.H.W.: Error reduction and convergence for an adaptive mixed finite element method. Math. Comp. 75, 1033–1042 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  12. Carstensen, C., Gallistl, D., Schedensack, M.: Quasi-optimal adaptive pseudostress approximation of the Stokes equations. SIAM J. Numer. Anal. 51, 1715–1734 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  13. Chen, L., Holst, M., Xu, J.: Convergence and optimality of adaptive mixed finite element methods. Math. Comp. 78, 35–53 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Becker, R., Mao, S., Shi, Z.: A convergent nonconforming adaptive finite element method with quasi-optimal complexity. SIAM J. Numer. Anal. 47, 4639–4659 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  15. Becker, R., Mao, S.: Quasi-optimality of adaptive nonconforming finite element methods for the Stokes equations. SIAM J. Numer. Anal. 49, 970–991 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  16. Liu, W., Yan, N.: A posteriori error analysis for convex distributed optimal control problems. Adv. Comp. Math. 15, 285–309 (2001)

    Article  MATH  Google Scholar 

  17. 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  MathSciNet  MATH  Google Scholar 

  18. Kohls, K., Rösch, A., Siebert, K.G.: A posteriori error analysis of optimal control problems with control constraints. SIAM J. Control. Optim. 52, 1832–1861 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  19. Liu, W., Yan, N.: A posteriori error estimates for convex boundary control problems. SIAM J. Numer. Anal. 39, 73–99 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  20. Liu, W., Yan, N.: A posteriori error estimates for optimal problems governed by Stokes equations. SIAM J. Numer. Anal. 40, 1850–1869 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  21. Chen, Y., Liu, W.: A posteriori error estimates for mixed finite element solutions of convex optimal control problems. J. Comput. Appl. Math. 211, 76–89 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  22. Chen, Y., Liu, L., Lu, Z.: A posteriori error estimates of mixed methods of quadratic optimal control probems. Numer. Func. Anal. Opt. 31, 1135–1157 (2010)

    Article  MATH  Google Scholar 

  23. Li, R., Liu, W., Ma, H., Tang, T.: Adaptive finite element approximation for distributed elliptic optimal control problems. SIAM J. Control. Optim. 41, 1321–1349 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  24. Ge, L., Liu, W., Yang, D.: l 2 norm equivalent a posteriori error estimate for a constrained optimal control problem. Inter. J. Numer. Anal. Model. 6, 335–353 (2009)

    MATH  Google Scholar 

  25. Ge, L., Liu, W., Yang, D.: Adaptive finite element approximation for a constrained optimal control problem via multi-meshes. J. Sci. Comput. 41, 238–255 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  26. Hintermtüller, M., Hoppe, R.H.W., Iliash, Y., Kieweg, M.: An a posteriori error analysis of adaptive finite element methods for distributed elliptic control problems with control constraints. ESIAM: COCV 14, 540–560 (2008)

    MathSciNet  MATH  Google Scholar 

  27. Lions, J.L.: Optimal control of systems governed by partial differential equations. Springer, Berlin (1971)

    Book  MATH  Google Scholar 

  28. Gaevskaya, A., Hoppe, R.H.W., Iliash, Y., Kieweg, M.: Convergence analysis of an adaptive finite element method for distributed control problems with control constraints. Internat. Ser. Numer. Math. 155, 47–68 (2007)

    MathSciNet  MATH  Google Scholar 

  29. Kohls, K., Rösch, A., Siebert, K.G.: Convergence of adaptive finite elements for control constrained optimal control problems Preprint-Nr.: SPP1253-153 (2013)

  30. Dai, X., Xu, J., Zhou, A.: Convergence and optimal complexity of adaptive finite element eigenvalue computations. Numer. Math. 110, 313–355 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  31. He, L., Zhou, A.: Convergence and complexity of adaptive finite element methods for elliptic partial differential equations. Inter. J. Numer. Anal. Model. 8, 1721–1743 (2011)

    MathSciNet  Google Scholar 

  32. Gong, W., Yan, N.: Adaptive finite element method for elliptic optimal control problems: convergence and optimality. Preprint arXiv:1505.01632(2015)

  33. Xu, J., Zhou, A.: Local and parallel finite element algorithms based on two-grid discretizations. Math. Comp. 69, 881–909 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  34. Verfürth, R.: A Review of a Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley-Teubner, Chichester (1996)

    MATH  Google Scholar 

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Acknowledgement

The authors thank the helps provided by Wei Gong in the aspect of numerical experiments and the anonymous referees for their valuable comments and suggestions. We note here that the second author is supported by National Science Foundation of China (11671157, 91430104) and the first author is supported by The Scientific Research Foundation of Graduate School of South China Normal University (2016lkxm21, 2016YN16).

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Authors and Affiliations

  1. School of Mathematical Science, South China Normal University, Guangzhou, 520631, Guangdong, People’s Republic of China

    Haitao Leng & Yanping Chen

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  1. Haitao Leng
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  2. Yanping Chen
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Correspondence to Yanping Chen.

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Communicated by: Long Chen

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Leng, H., Chen, Y. Convergence and quasi-optimality of an adaptive finite element method for optimal control problems with integral control constraint. Adv Comput Math 44, 367–394 (2018). https://doi.org/10.1007/s10444-017-9546-8

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  • DOI: https://doi.org/10.1007/s10444-017-9546-8

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