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Higher Order Triangular Mixed Finite Element Methods for Semilinear Quadratic Optimal Control Problems

Published online by Cambridge University Press:  28 May 2015

Kang Deng ,
Yanping Chen  and
Zuliang Lu
Kang Deng*
Affiliation:
School of Mathematical Sciences, Hunan University of Science and Technology, Xiangtan 411201, P.R. China
Yanping Chen*
Affiliation:
School of Mathematical Sciences, South China Normal University, Guangzhou 510631, P.R. China
Zuliang Lu*
Affiliation:
College of Mathematics and Computer Sciences, Chongqing Three Gorges University, Chongqing 404000, P.R. China College of Civil Engineering and Mechanics, Xiangtan University, Xiangtan 411105, P.R. China
*
Corresponding author.Email address:kdeng@hnust.edu.cn
Corresponding author.Email address:yanpingchen@scnu.edu.cn
Corresponding author.Email address:zulianglux@126.com
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Abstract

In this paper, we investigate a priori error estimates for the quadratic optimal control problems governed by semilinear elliptic partial differential equations using higher order triangular mixed finite element methods. The state and the co-state are approximated by the order k Raviart-Thomas mixed finite element spaces and the control is approximated by piecewise polynomials of order k (k ≥ 0). A priori error estimates for the mixed finite element approximation of semilinear control problems are obtained. Finally, we present some numerical examples which confirm our theoretical results.

Keywords

49J2065N30a priori error estimatessemilinear optimal control problemshigher order triangular elementsmixed finite element methods

Information

Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 4 , Issue 2 , May 2011 , pp. 180 - 196
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]Brezzi, F., On the existence, uniqueness and approximation of saddle-point problems arisingfrom Lagrangian multipliers, RAIRO Anal. Numer., 2 (1974), pp. 129–151.Google Scholar
[4]Brezzi, F. and Fortin, M., Mixed and hybrid finite element methods, Springer, Berlin, 1991.CrossRefGoogle Scholar
[5]Bonnans, J. F., Second-order analysis for control constrained optimal control problems of semi-linear elliptic systems, Appl. Math. Optim., 38 (1998), pp. 303–325.CrossRefGoogle Scholar
[6]Chen, Y., Superconvergence of optimal control problems by rectangular mixed finite element methods, Math. Comp., 77 (2008), pp. 1269–1291.CrossRefGoogle Scholar
[7]Chen, Y., Superconvergence of quadratic optimal control problems by triangular mixed finite elements, Internat. J. Numer. Methods in Engineering, 75 (2008), pp. 881–898.CrossRefGoogle Scholar
[8]Chen, Y. and Liu, W. B., Error estimates and superconvergence of mixed finite elements for quadratic optimal control, Internat. J. Numer. Anal. Modeling, 3 (2006), pp. 311–321.Google Scholar
[9]Chen, Y. and Lu, Z., Error estimates of fully discrete mixed finite element methods for semilinear quadratic parabolic optimal control problems, Comput. Methods Appl. Mech. Engrg., 199 (2010), pp. 1415–1423.CrossRefGoogle Scholar
[10]Clement, P., Approximation by finite element funtions using local regularization, RAIRO Ser. Rougr Anal. Numer., 2 (1995), pp. 77–84.Google Scholar
[11]Falk, F. S., Approximation of a class of optimal control problems with order of convergence estimates, J. Math. Anal. Appl., 44 (1973), pp. 28–47.CrossRefGoogle Scholar
[12]Geveci, T., On the approximation of the solution of an optimal control problem governed by an elliptic equation, RAIRO: Numer. Anal., 13 (1979), pp. 313–328.Google Scholar
[13]Gilbarg, D. and Truninger, N., Elliptic partial differential equations of second order, Springerverlag, Berlin, 1977.CrossRefGoogle Scholar
[14]Grisvard, P., Elliptic problems in nonsmooth domains, Pitman, London, 1985.Google Scholar
[15]Gunzburger, M. D. and Hou, S. L., Finite dimensional approximation of a class of constrained nonlinear control problems, SIAM J. Control Optim., 34 (1996), pp. 1001–1043.CrossRefGoogle Scholar
[16]Huang, Y., Li, R., Liu, W. B. and Yan, N. N., Adaptive multi-mesh finite element approximation for constrained optimal control problems, SIAM J. Control and Optim., in press.Google Scholar
[17]Li, R., Ma, H., Liu, W. B., and Tang, T., Adaptive finite element approximation for distributed elliptic optimal control problems, SIAM J. Control and Optimization, 41 (2002), pp. 1321–1349.CrossRefGoogle Scholar
[18]Li, R. and Liu, W. B., http://circus.math.pku.edu.cn/AFEPack.Google Scholar
[19]Lions, J. L., Optimal control of systems governed by partial differential equtions, Springer, Berlin, 1971.CrossRefGoogle Scholar
[20]Liu, W. B. and Yan, N. N., A posteriori error estimates for distributed convex optimal control problems, Numer. Math., 101 (2005), pp. 1–27.Google Scholar
[21]Liu, W. B. and Yan, N. N., A posteriori error estimates for control problems governed by nonlinear elliptic equation, Adv. Comp. Math., 15 (2001), pp. 285–309.CrossRefGoogle Scholar
[22]Lu, Z. and Chen, Y, A posteriori error estimates of triangular mixed finite element methods for semilinear optimal control problems, Adv. Appl. Math. Mech., 1 (2009), pp. 242–256.Google Scholar
[23]LU, Z. and Hen, Y C, L∞-error estimates of triangular mixed finite element methods for optimal control problem govern by semilinear elliptic equation, Numer. Anal. Appl., 12 (2009), pp. 74–86.CrossRefGoogle Scholar
[24]Lu, Z., Chen, Y. and Zhang, H., A priori error estimates of mixed finite element methods for nonlinear quadratic optimal control problems, Lobachevskii J. Math., 29 (2008), pp. 164–174.CrossRefGoogle Scholar
[25]Miliner, F. A., Mixed finite element methods for quasilinear second-order elliptic problems, Math. Comp., 44 (1985), pp. 303–320.CrossRefGoogle Scholar
[26]Kwon, Y. and Milner, F. A., L∞ -error estimates for mixed methods for semilinear second-order elliptic equations, SIAM J. Numer. Anal., 25 (1988), pp. 46–53.CrossRefGoogle Scholar
[27]Xing, X. and Chen, Y., Error estimates of mixed methods for optimal control problems governed by parabolic equations, Internat. J. Numer. Methods Engrg., 75 (2008), pp. 735–754.CrossRefGoogle Scholar