Hostname: page-component-848d4c4894-5nwft Total loading time: 0 Render date: 2024-04-30T15:26:07.962Z Has data issue: false hasContentIssue false
  • English
  • Français

Recovery Type A Posteriori Error Estimates of Fully Discrete Finite Element Methods for General Convex Parabolic Optimal Control Problems

Published online by Cambridge University Press:  28 May 2015

Yuelong Tang  and
Yanping Chen
Yuelong Tang*
Affiliation:
Hunan Key Laboratory for Computation and Simulation in Science and Engineering, School of Mathematics and Computational Science, Xiangtan University, Xiangtan 411105, Hunan, China
Yanping Chen*
Affiliation:
School of Mathematical Sciences, South China Normal University, Guangzhou 510631, Guangdong, China
*
Corresponding author.Email address:tangyuelonga@163.com
Corresponding author.Email address:yanpingchen@scnu.edu.en
Get access

Abstract

This paper is concerned with recovery type a posteriori error estimates of fully discrete finite element approximation for general convex parabolic optimal control problems with pointwise control constraints. The time discretization is based on the backward Euler method. The state and the adjoint state are approximated by piece-wise linear functions and the control is approximated by piecewise constant functions. We derive the superconvergence properties of finite element solutions. By using the superconvergence results, we obtain recovery type a posteriori error estimates. Some numerical examples are presented to verify the theoretical results.

Keywords

35B3749J2065N30General convex optimal control problemsfully discrete finite element approximationa posteriori error estimatessuperconvergencerecovery operator
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 5 , Issue 4 , November 2012 , pp. 573 - 591
Copyright
Copyright © Global Science Press Limited 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]Ainsworth, M. and Oden, J.T., A Posteriori Error Estimation in Finite Element Analysis, Wiley Interscience, New York, 2000.CrossRefGoogle Scholar
[2]Babuška, I., Strouboulis, T., Upadhyay, C.S. and Gangaraj, S.K., A posteriori estimation and adaptive control of the pollution error in the h-version of the finite element method, Int. J. Numer. Meth. Eng., 38(24) (1995), pp. 4207–4235.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(1) (2000), pp. 113–132.CrossRefGoogle Scholar
[4]Becker, R., Meidner, D. and Vexler, B., Efficient numerical solution of parabolic optimization problems by finite element methods, Optim. Meth. Soft., 22(5) (2007), pp. 813–833.CrossRefGoogle Scholar
[5]Brandts, J., Superconvergence and a posteriori error estimation for triangular mixed finite elements, Numer. Math., 68 (1994), pp. 311–324.CrossRefGoogle Scholar
[6]Carstensen, C. and Verfürth, R., Edge residuals donminate a posteriori error estimates for low order finite element methods, SIAM J. Numer. Anal., 36(5) (1999), pp. 1571–1587.CrossRefGoogle Scholar
[7]Chen, C. and Huang, Y., High Accuracy Theory of Finite Element Methods, Hunan Science and Technology Press, Hunan, 1995.Google Scholar
[8]Chen, Y., Superconvergence of mixed finite element methods for optimal control problems, Math. Comput., 77 (2008), pp. 1269–1291.CrossRefGoogle Scholar
[9]Chen, Y., Superconvergence of quadratic optimal control problems by triangular mixed finite elements, Int. J. Numer. Meth. Eng., 75(8) (2008), pp. 881–898.CrossRefGoogle Scholar
[10]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
[11]Chen, Y., Fu, Y., Liu, H., Dai, Y. and Wei, H., Recovery a posteriori error estimates for general convex elliptic optimal control problems subject to pointwise control constraints, J. Comput. Math., 27(4) (2009), pp. 543–560.CrossRefGoogle Scholar
[12]Chen, Y., Huang, Y., Liu, W. and Yan, N., Error estimates and superconvergence of mixed finite element methods for convex optimal control problems, J. Sci. Comput., 42 (2010), pp. 382–403.Google Scholar
[13]Chen, Y. and Liu, W., Error estimates and superconvergence of mixed finite element for quadratic optimal control, Int. J. Numer. Anal. Model., 3 (2006), pp. 311–321.Google Scholar
[14]Chen, Y. and Tang, Y., Superconvergence analysis of fully discrete finite element methods for semilinear parabolic optimal control problems, Front. Math. China, appear.Google Scholar
[15]Ciarlet, P. G., The Finite Element Method for Elliptic Problems, North-Holland, Amstterdam, 1978.Google Scholar
[16]Fu, H. and Rui, H., A priori error estimates for optimal control problems governed by transient advection-diffusion equationg, J. Sci. Comput., 38 (2009), pp. 290–315.CrossRefGoogle Scholar
[17]Huang, M., Numerical Methods for Evolution Equations, Science Press, Beijing, 2004.Google Scholar
[18]Huang, Y. and Xu, J., Superconvergence of quadratic finite elements on mildly structured grids, Math. Comp., 77 (2008), pp. 1253–1268.CrossRefGoogle Scholar
[19]Knowles, G., Finite element approxiamtion of parabolic time optimal control problems, SIAM J. Control Optim., 20(3) (1982), pp. 414–427.CrossRefGoogle Scholar
[20]Kufner, A., John, O. and Fuck, S., Function Spaces, Nordhoff, Leiden, The Netherlands, 1997.Google Scholar
[21]Li, R., Liu, W. and Yan, N., A posteriori error estimates of recovery type for distributed convex optimal control problems, J. Sci. Comput., 33 (2007), pp. 155–182.CrossRefGoogle Scholar
[22]Lin, Q. and Zhu, Q., The Preprocessing and Postprocessing for the Finite Element Method, Shanghai Scientific and Technical Publishers, Shanghai, 1994.Google Scholar
[23]Lions, J., Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, Berlin, 1971.CrossRefGoogle Scholar
[24]Lions, J. and Magenes, E., Non Homogeneous Boundary Value Problems and Applications, Springer-verlag, Berlin, 1972.Google Scholar
[25]Liu, H. and Yan, N., Recovery type superconvergence and a posteriori error estimates for control problems governed by Stokes equations, J. Comput. Appl. Math., 209 (2007), pp. 187–207.CrossRefGoogle Scholar
[26]Liu, W. and Yan, N., A posteriori error estimates for distributed convex optimal control problems, Adv. Comput. Math. 15 (2001) 285–309.CrossRefGoogle Scholar
[27]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
[28]Liu, W. and Yan, N., Adaptive Finite Element Methods for Optimal Control Governed by PDEs, Science Press, Beijing, 2008.Google Scholar
[29]Meidner, 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(3) (2008), pp. 1150–1177.Google Scholar
[30]Meidner, 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(3) (2008), pp. 1301–1329.Google Scholar
[31]Meyer, C. and Rösch, A., Superconvergence properties of optimal control problems, SIAM J. Control Optim., 43(3) (2004), pp. 970–985.CrossRefGoogle Scholar
[32]Neittaanmaki, P. and Tiba, D., Optimal Control of Nonlinear Parabolic Systems: Theory, Algorithms and Applications, Dekker, New York, 1994.Google Scholar
[33]Tiba, D., Lectures on The Optimal Control of Elliptic Equations, University of Jyvaskyla Press, Finland, 1995.Google Scholar
[34]Xiong, C. and Li, Y., A posteriori error estimates for optimal distributed control governed by the evolution equations, Appl. Numer. Math., 61 (2011), pp. 181–200.CrossRefGoogle Scholar
[35]Yang, D., Chang, Y. and Liu, W., A priori error estimate and superconvergence annalysis for an opitmal control problem of bilinear type, J. Comput. Math., 26(4) (2008), pp. 471–487.Google Scholar
[36]Zhou, J., Chen, Y. and Dai, Y., Superconvergence of triangular mixed finite elements for optimal control problems with an integral constraint, Appl. Math. Comput., 217 (2010), pp. 2057–2066.Google Scholar
[37]Zienkiwicz, O. C. and Zhu, J.Z., The superconvergence patch recovery and a posteriori error estimates, Int. J. Numer. Meth. Eng., 33 (1992), pp. 1331–1382.Google Scholar
[38]Zienkiwicz, O. C. and Zhu, J.Z., The superconvergence patch recovery (SPR) and adaptive finite element refinement, Comput. Meth. Appl. Math., 101 (1992), pp. 207–224.Google Scholar