Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-22T21:55:29.587Z Has data issue: false hasContentIssue false
  • English
  • Français

A Posteriori Error Estimates of Lowest Order Raviart-Thomas Mixed Finite Element Methods for Bilinear Optimal Control Problems

Published online by Cambridge University Press:  28 May 2015

Zuliang Lu ,
Yanping Chen  and
Weishan Zheng
Zuliang Lu*
Affiliation:
School of Mathematics and Statistics, Chongqing Three Gorges University, Chongqing 404000 and College of Civil Engineering and Mechanics, Xiangtan University, Xiangtan 411105, P.R. China.
Yanping Chen*
Affiliation:
School of Mathematical Sciences, South China Normal University, Guangzhou 510631, P.R. China
Weishan Zheng*
Affiliation:
School of Mathematical Sciences, South China Normal University, Guangzhou 510631, P.R. China
*
Corresponding author. Email: [email protected]
Corresponding author. Email: [email protected]
Corresponding author. Email: [email protected]
Get access

Abstract

A Raviart-Thomas mixed finite element discretization for general bilinear optimal control problems is discussed. The state and co-state are approximated by lowest order Raviart-Thomas mixed finite element spaces, and the control is discretized by piecewise constant functions. A posteriori error estimates are derived for both the coupled state and the control solutions, and the error estimators can be used to construct more efficient adaptive finite element approximations for bilinear optimal control problems. An adaptive algorithm to guide the mesh refinement is also provided. Finally, we present a numerical example to demonstrate our theoretical results.

Keywords

49J2065N30Bilinear optimal control problemslowest order Raviart-Thomas mixed finite element methodsa posteriori error estimatesadaptive algorithm
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 2 , Issue 2 , May 2012 , pp. 108 - 125
Copyright
Copyright © Global-Science Press 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Casas, E., Mateos, M. and Tröltzsch, F., Necessary and sufficient optimality conditions for optimization problems in function spaces and applications to control theory, ESIAM Proceedings, 13 (2003), pp. 18–30.Google Scholar
[2]Casas, E. and Tröltzsch, F., Second order necessary and sufficient optimality conditions for optimization problems and applications to control theory, SIAM J. Optim., 13 (2002), pp. 406–431.CrossRefGoogle Scholar
[3]Babuska, I. and Strouboulis, T., The Finite Element Method and Its Reliability, Oxford University Press, Oxford, 2001.Google Scholar
[4]Brezzi, F. and Fortin, M., Mixed and Hybrid Finite Element Methods, Springer, Berlin, 1991.Google Scholar
[5]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.Google Scholar
[6]Becker, R. and Vexler, B., Optimal control of the convection-diffusion equation using stabilized finite element methods, Numer. Math., 106 (2007), pp. 347–367.Google Scholar
[7]Brunner, H. and Yan, N., Finite element methods for optimal control problems governed by integral equations and integro-differential equations, Appl. Numer. Math., 47 (2003), pp. 173–187.Google Scholar
[8]Carstensen, C., A posteriori error estimate for the mixed finite element method, Math. Comp., 66 (1997), pp. 465–476.Google Scholar
[9]Chen, Y., Superconvergence of mixed finite element methods for optimal control problems, Math. Comp., 77 (2008), pp. 1269–1291.CrossRefGoogle Scholar
[10]Chen, Y. and Liu, W., 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
[11]Chen, Y. and Lu, Z., Error estimates for parabolic optimal control problem by fully discrete mixed finite element methods, Finite Elem. Anal. Des., 46 (2010), pp. 957–965.Google Scholar
[12]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.Google Scholar
[13]Tao, T., Yan, N., and Liu, W., Adaptive finite element methods for the identification of distributed parameters in elliptic equation, Adv. Comput. Math., 29 (2008), pp. 27–53.Google Scholar
[14]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
[15]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
[16]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
[17]Huang, Y., Li, R., Liu, W., and Yan, N., Adaptive multi-mesh finite element approximation for constrained optimal control problems, SIAM J. Control Optim., in press.Google Scholar
[18]Shi, Z. and Wang, M., Finite Element Methods, Science Press, Beijing, 2010.Google Scholar
[19]Li, R., Liu, W., Ma, H. and Tang, T., Adaptive finite element approximation for distributed elliptic optimal control problems, SIAM J. Control Optim., 41 (2002), pp. 1321–1349.Google Scholar
[20]Liu, W. and Yan, N., A posteriori error estimates for distributed convex optimal control problems, Numer. Math., 101 (2005), pp. 1–27.Google Scholar
[21]Liu, W. and Yan, N., A posteriori error estimates for control problems governed by nonlinear elliptic equation, Adv. Comp. Math., 15 (2001), pp. 285–309.Google 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 Chen, Y., 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]Miliner, F. A., Mixed finite element methods for quasilinear second-order elliptic problems, Math. Comp., 44 (1985), pp. 303–320.Google Scholar
[25]Raviart, P. A. and Thomas, J. M., A mixed finite element method for 2nd order elliptic problems, in Mathematical Aspects of Finite Element Methods, (Eds. Galligani, I. and Magenes, E.), Lecture Notes in Math., 606 (Springer, Berlin, 1977), pp. 292–315.Google Scholar
[26]Verfurth, R., A Review of a Posteriori Error Estimation and Adaptive Mesh-Refinment Techniques, Wiley-Teubner, 1996.Google Scholar