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A Posteriori Error Estimates of Mixed Methods for Quadratic Optimal Control Problems Governed by Parabolic Equations

Published online by Cambridge University Press:  28 May 2015

Tianliang Hou ,
Yanping Chen  and
Yunqing Huang
Tianliang Hou*
Affiliation:
Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Department of Mathematics, Xiangtan University, Xiangtan 411105, Hunan, China
Yanping Chen*
Affiliation:
School of Mathematical Sciences, South China Normal University, Guangzhou 510631, Guangdong, China
Yunqing Huang*
Affiliation:
Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Department of Mathematics, Xiangtan University, Xiangtan 411105, Hunan, China
*
Corresponding author.Email address:[email protected]
Corresponding author.Email address:[email protected]
Corresponding author.Email address:[email protected]
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Abstract

In this paper, we discuss the a posteriori error estimates of the mixed finite element method for quadratic optimal control problems governed by linear parabolic equations. The state and the co-state are discretized by the high order Raviart-Thomas mixed finite element spaces and the control is approximated by piecewise constant functions. We derive a posteriori error estimates for both the state and the control approximation. Such estimates, which are apparently not available in the literature, are an important step towards developing reliable adaptive mixed finite element approximation schemes for the control problem.

Keywords

49J2065N30A posteriori error estimatesquadratic optimal control problemsparabolic equationsmixed finite element methods
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 4 , Issue 4 , November 2011 , pp. 439 - 458
Copyright
Copyright © Global Science Press Limited 2011

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References

[1]Arada, N., Casas, E. and Tröltzsch, F., Error estimates for the numerical approximation of a semilinear elliptic control problem, Comp. Optim. Appl., 23 (2002), pp. 201–229.CrossRefGoogle Scholar
[2]Babuska, I. and Strouboulis, T., The Finite Element Method and its Reliability, Oxford University press, Oxford, 2001.CrossRefGoogle Scholar
[3]Becker, R., Kapp, H. and Rannacher, R., Adaptive finite element methods for optimal control of partial differential equations: Basic concept, SIAM J. Control Optim., 39 (2000), pp. 113–132.CrossRefGoogle Scholar
[4]Brunner, H. and Yan, N., Finite element methods for optimal control problems governed by integral equations and integro-differential equations, Numer. Math., 101 (2005), pp. 1–27.CrossRefGoogle Scholar
[5]Brezzi, F. and Fortin, M., Mixed and hybrid finite element methods, Springer-Verlag., 95 (1991), pp. 65–187.Google Scholar
[6]Chen, Y., Superconvergence of quadratic optimal control problems by triangular mixed finite elements, Inter. J. Numer. Meths. Eng., 75 (2008), pp. 881–898.CrossRefGoogle Scholar
[7]Chen, Y., Huang, Y., Liu, W. B. and Yan, N. N., Error estimates and superconvergence of mixed finite element methods for convex optimal control problems, J. Sci. Comput., 42 (2009), pp. 382–403.CrossRefGoogle Scholar
[8]Chen, Y. and Liu, W. B., A posteriori error estimates for mixed finite element solutions of convex optimal control problems, J. Comp. Appl. Math., 211 (2008), pp. 76–89.CrossRefGoogle Scholar
[9]Chen, Y., Liu, L. and Lu, Z., A posteriori error estimates of mixed methods for parabolic optimal control problems, Numer. Funct. Anal. Optim., 31 (2010), pp. 1135–1157.CrossRefGoogle Scholar
[10]Ciarlet, P. G., The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978.Google Scholar
[11]Carstensen, C., A posteriori error estimate for the mixed finite element method, Math. Comp., 66 (1997), pp. 465–476.CrossRefGoogle Scholar
[12]Eriksson, K. and Johnson, C., Adaptive finite elements methods for parabolic problems I: a linear model problem, SIAM J. Numer. Anal., 28 (1991), pp. 43–77.CrossRefGoogle Scholar
[13]Gong, W. and Yan, N., A posteriori error estimate for boundary control problems governed by the parabolic partial differential equations, J. Comput. Math., 27 (2009), pp. 68–88.Google Scholar
[14]Hou, L. and Turner, J. C., Analysis and finite element approximation of an optimal control problem in electrochemistry with current density controls, Numer. Math., 71 (1995), pp. 289–315.CrossRefGoogle Scholar
[15]Haslinger, J. and Neittaanmaki, P., Finite element approximation for optimal shape design, John Wiley and Sons, Chichester, 1989 (in UK).Google Scholar
[16]Hoppe, R. H. W., Iliash, Y., Iyyunni, C. and Sweilam, N. H., A posteriori error estimates for adaptive finite element discretizations of boundary control problems, J. Numer. Math., 14 (2006), pp. 57–82.CrossRefGoogle Scholar
[17]Knowles, G., Finite element approximation of parabolic time optimal control problems, SIAM J. Control Optim., 20 (1982), pp. 414–427.CrossRefGoogle Scholar
[18]Lions, J. L., Optimal Control of Systems Governed by Partial Differential Equations, Springer, Berlin, 1971.CrossRefGoogle Scholar
[19]Liu, W., Ma, H., Tang, T. and Yan, N., A posteriori error estimates for discontinuous Galerkin time-stepping method for optimal control problems governed by parabolic equations, SIAM J. Numer. Anal., 42 (2004), pp. 1032–1061.CrossRefGoogle Scholar
[20]Liu, W. and Yan, N., A posteriori error estimates for convex boundary control problems, SIAM J. Numer. Anal., 39 (2001), pp. 73–99.CrossRefGoogle Scholar
[21]Liu, W. and Yan, N., A posteriori error analysis for convex distributed optimal control problems, Adv. Comp. Math., 15 (2001), pp. 285–309.CrossRefGoogle Scholar
[22]Liu, W. and Yan, N., A posteriori error estimates for optimal control problems governed by Stokes equations, SIAM J. Numer. Anal., 40 (2003), pp. 1850–1869.CrossRefGoogle Scholar
[23]Liu, W. and Yan, N., A posteriori error estimates for optimal control problems governed by parabolic equations, Numer. Math., 93 (2003), pp. 497–521.CrossRefGoogle Scholar
[24]Li, R., Liu, W., Ma, H. and Tang, T., Adaptive finite element approximation of elliptic control problems, SIAM J. Control Optim., 41 (2002), pp. 1321–1349.CrossRefGoogle Scholar
[25]Mcknight, R. and Bosarge, W. Jr., The Ritz-Galerkin procedure for parabolic control problems, SIAM J. Control Optim., 11 (1973), pp. 510–524.CrossRefGoogle Scholar
[26]Neittaanmaki, P. and Tiba, D., Optimal Control of Nonlinear Parabolic Systems: Theory, Algorithms and Applications, M. Dekker, New York, 1994.Google Scholar
[27]Scott, L. R. and Zhang, S., Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comput., 54 (1990), pp. 483–493.CrossRefGoogle Scholar
[28]Tiba, D., Lectures on the optimal control of elliptic problems, University of Jyvaskyla Press, Jyvaskyla, 1995.Google Scholar
[29]Thomée, V., Galerkin Finite Element Methods for Parabolic Problems, Springer-Verlag, 1997.CrossRefGoogle Scholar
[30]Tröltzsch, F., Semidiscrete Ritz-Galerkin approximation of nonlinear parabolic boundary control problems-strong convergence of optimal control, Appl. Math. Optim., 29 (1994), pp. 309–329.CrossRefGoogle Scholar