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Error Estimates and Superconvergence of Mixed Finite Element Methods for Optimal Control Problems with Low Regularity

Published online by Cambridge University Press:  03 June 2015

Yanping Chen ,
Tianliang Hou  and
Weishan Zheng
Yanping Chen*
Affiliation:
School of Mathematical Sciences, South China Normal University, Guangzhou 510631, Guangdong, China
Tianliang Hou*
Affiliation:
Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Department of Mathematics, Xiangtan University, Xiangtan 411105, Hunan, China
Weishan Zheng*
Affiliation:
School of Mathematical Sciences, South China Normal University, Guangzhou 510631, Guangdong, China
*
Corresponding author. URL: http: //math.xtu.edu.cn/myphp/math/ypchen/index.htm, Email: yanpingchen@scnu.edu.cn
Email: haohan_9@126.com
Email: htlchb@163.com
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Abstract

In this paper, we investigate the error estimates and superconvergence property of mixed finite element methods for elliptic optimal control problems. The state and co-state are approximated by the lowest order Raviart-Thomas mixed finite element spaces and the control variable is approximated by piecewise constant functions. We derive L2 and L∞-error estimates for the control variable. Moreover, using a recovery operator, we also derive some superconvergence results for the control variable. Finally, a numerical example is given to demonstrate the theoretical results.

Keywords

49J2065N30Elliptic equationsoptimal control problemssuperconvergenceerror estimatesmixed finite element methods
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 4 , Issue 6 , December 2012 , pp. 751 - 768
Copyright
Copyright © Global-Science Press 2012

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References

[1]Bonnans, J. F. and Casas, E., An extension of Pontryagin’s principle for state constrained optimal control of semilinear elliptic eqnation and variational inequalities, SIAM J. Control Optim., 33 (1995), pp. 274–298.Google Scholar
[2]Brezzi, F. and Fortin, M., Mixed and Hybrid Finite Element Methods, Springer-Verlag, New York, 1991.Google Scholar
[3]Chen, Y., Superconvergence of mixed finite element methods for optimal control problems, Math. Comput., 77 (2008), pp. 1269–1291.Google Scholar
[4]Chen, Y., Superconvergence of quadratic optimal control problems by triangular mixed finite elements, Int. J. Numer. Meths. Eng., 75(8) (2008), pp. 881–898.Google Scholar
[5]Chen, Y. and Dai, Y., Superconvergence for optimal control problems governed by semi-linear elliptic equations, J. Sci. Comput., 39 (2009), pp. 206–221.CrossRefGoogle Scholar
[6]Chen, Y.-P., Dai, L. and Lu, Z.-L., Superconvergence of rectangular mixed finite element methods for constrained optimal control problem, Adv. Appl. Math. Mech., 2 (2010), pp. 56–75.CrossRefGoogle Scholar
[7]Chen, Y.-P.Hou, T.-L., Superconvergence and L∞-error estimates of RT1 mixed methods for semilinear elliptic control problems with an integral constraint, Numer. Math. Theor. Meth. Appl., 5 (2012), pp. 423–446.Google Scholar
[8]Chen, Y., Huang, Y., Liu, W. B. and Yan, N., Error estimates and superconvergence of mixed finite element methods for convex optimal control problems, J. Sci. Comput., 42(3) (2009), pp. 382–403.Google Scholar
[9]Ciarlet, P. G., The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978.Google Scholar
[10]Douglas, J. and Roberts, J. E., Global estimates for mixed finite element methods for second order elliptic equations, Math. Comput., 44 (1985), pp. 39–52.Google Scholar
[11]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.Google Scholar
[12]Grisvard, P., Elliptic Problems in Nonsmooth Domains, Pitman, Boston-London Melbourne, 1985.Google Scholar
[13]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.Google Scholar
[14]Huang, Y.-Q., Qin, H.-F., Wang, D.-S. and Du, Q., Convergent adaptive finite element method based on Centroidal Voronoi Tessellations and superconvergence, Commun. Comput. Phys., 10 (2011), pp. 339–370.Google Scholar
[15]Knowles, G., Finite element approximation of parabolic time optimal control problems, SIAM J. Control Optim., 20 (1982), pp. 414–427.Google Scholar
[16]Li, R. and Liu, W., .Google Scholar
[17]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.Google Scholar
[18]Li, R., Liu, W. B. and Yan, N., A posteriori error estimates of recovery type for distributed convex optimal control problems, J. Sci. Comput., 41(5) (2002), pp. 1321–1349.Google Scholar
[19]Lions, J. L., Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, Berlin, 1971.Google Scholar
[20]Liu, T., Yan, N. and Zhang, S.H., Richardson extrapolation and defect correction of finite element methods for optimal control problem, J. Comput. Math., 28 (2010), pp. 55–71.Google Scholar
[21]Meyer, C. and Rösch, A., Superconvergence properties of optimal control problems, SIAM J. Control Optim., 43(3) (2004), pp. 970–985.CrossRefGoogle Scholar
[22]Meyer, C. and Rösch, A., L∞-error estimates for approximated optimal control problems, SIAM J. Control Optim., 44 (2005), pp. 1636–1649.Google Scholar
[23]Meider, D. and Vexler, B., A priori error estimates for space-time finite element discretization of parabolic optimal control problems part I: problems without control constraints, SIAM J. Control Optim., 47 (2008), pp. 1150–1177.Google Scholar
[24]Meider, D. and Vexler, B., A priori error estimates for space-time finite element discretization of parabolic optimal control problems part II: problems with control constraints, SIAM J. Control Optim., 47 (2008), pp. 1301–1329.Google Scholar
[25]Mckinght, R. S. and Borsarge, J., The Ritz-Galerkin procedure for parabolic control problems, SIAM J. Control Optim., 11 (1973), pp. 510–542.Google Scholar
[26]Niu, H.-F. and Yang, D.-P., Finite element analysis of optimal control problem governed by Stokes equations with L2-norm state-constraints, J. Comput. Math., 29 (2011), pp. 589–604.Google Scholar
[27]Raviart, P. A. and Thomas, J. M., A mixed finite element method for 2nd order elliptic problems, aspecs of the finite element method, Lecture Notes In Math, Springer, Berlin, 606 (1977), pp. 292–315.Google Scholar
[28]Ryu, S., Kim, S. D. and H.-C. LEE, First-order system least-squares methods for a flux control problem by the Stokes flow, Commun. Comput. Phys., 7 (2010), pp. 738–758.Google Scholar
[29]Tang, Y.-L. and Chen, Y.-P., Recovery type a posteriori error estimates of fully discrete finite element methods for general convex parabolic optimal control problems, Numer. Math. Theor. Meth. Appl., 5 (2012), pp. 573–591.Google Scholar
[30]Xing, X.-Q. and Chen, Y.-P., Superconvergence of parabolic optimal control problems, Adv. Appl. Math. Mech., 3 (2011), pp. 401–419.Google Scholar
[31]Yang, D., Chang, Y. and Liu, W., A priori error estimates and superconvergence analysis for an optimal control problems of bilinear type, J. Comput. Math., 4 (2008), pp. 471–487.Google Scholar
[32]Yan, N., Superconvergence analysis and a posteriori error estimation of a finite element method for an optimal control problem governed by integral equations, Appl. Math., 54 (2009), pp. 267–283.Google Scholar