Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-28T00:13:40.643Z Has data issue: false hasContentIssue false
  • English
  • Français

Superconvergence of Finite Element Methods for Optimal Control Problems Governed by Parabolic Equations with Time-Dependent Coefficients

Published online by Cambridge University Press:  28 May 2015

Yuelong Tang  and
Yanping Chen
Yuelong Tang
Affiliation:
Department of Mathematics and Computational Science, Hunan University of Science and Engineering, Yongzhou 425100, Hunan, China
Yanping Chen*
Affiliation:
School of Mathematical Sciences, South China Normal University, Guangzhou 510631, Guangdong, China
*
Corresponding author. Email Address: [email protected]
Get access

Abstract

In this article, a fully discrete finite element approximation is investigated for constrained parabolic optimal control problems with time-dependent coefficients. The spatial discretisation invokes finite elements, and the time discretisation a nonstandard backward Euler method. On introducing some appropriate intermediate variables and noting properties of the L2 projection and the elliptic projection, we derive the superconvergence for the control, the state and the adjoint state. Finally, we discuss some numerical experiments that illustrate our theoretical results.

Keywords

35B3749J2065N30Superconvergencefinite element methodsoptimal control problemsparabolic equationsinterpolate operator
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 3 , Issue 3 , August 2013 , pp. 209 - 227
Copyright
Copyright © Global-Science Press 2013

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. and Tröltzsch, F., Second-order necessary and sufficient optimality conditions for optimisation problems and applications to control theory, SIAM J. Optim. 13, 406–431 (2002).Google Scholar
[2]Chen, Y., Superconvergence of mixed finite element methods for optimal control problems, Math. Comp. 77, 1269–1291 (2008).CrossRefGoogle Scholar
[3]Chen, Y. and Dai, Y., Superconvergence for optimal control problems governed by semi-linear elliptic equations, J. Sci. Comput. 39, 206–221 (2009).CrossRefGoogle Scholar
[4]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, 382–403 (2010).CrossRefGoogle Scholar
[5]Chen, Y. and Liu, W., A posteriori error estimates for mixed finite element solutions of convex optimal control problems, J. Comput. Appl. Math. 211, 76–89 (2008).CrossRefGoogle Scholar
[6]Chen, Y. and Tang, Y., Numerical methods for constrained elliptic optimal control problems with rapidly oscillating coefficients, East Asian J. Appl. Math. 1, 235–247 (2011).Google Scholar
[7]Fu, H. and Rui, H., A priori error estimates for optimal control problems governed by transient advection-diffusion equationg, J. Sci. Comput. 38, 290–315 (2009).Google Scholar
[8]Glowinski, R., Rodin, E.Y. and Zienkiewicz, O.C., Energy Methods in Finite Element Analysis, Wiley Interscience, New York (1979).Google Scholar
[9]Hinze, M., A variational discretisation concept in cotrol constrained optimisation: The linear-quadratic case, Comput. Optim. Appl. 30 45–63 (2005).CrossRefGoogle Scholar
[10]Hinze, M., Yan, N. and Zhou, Z., Variational discretisation for optimal control governed by convection dominated diffusion equations, J. Comput. Math. 27, 237–253 (2009).Google Scholar
[11]Huang, M., Numerical Methods for Evolution Equations, Science Press, Beijing (2004).Google Scholar
[12]Li, R., Liu, W., Ma, H. and Tang, T., Adaptive finite element approximation for distributed elliptic optimal control problems, SIAM J. Control Optim. 41, 1321–1349 (2002).CrossRefGoogle Scholar
[13]Li, R., Liu, W. and Yan, N., A posteriori error estimates of recovery type for distributed convex optimal control problems, J. Sci. Comput. 33, 155–182 (2007).Google Scholar
[14]Lin, Q. and Zhu, Q., The Preprocessing and Postprocessing for the Finite Element Method, Shanghai Scientific and Technical Publishers, Shanghai (1994).Google Scholar
[15]Lions, J., Optimal Control of Systems Governed by Partial Differential Equations, Springer Verlag, Berlin (1971).Google Scholar
[16]Lions, J. and Magenes, E., Non Homogeneous Boundary Value Problems and Applications, Springer Verlag, Berlin (1972).Google Scholar
[17]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, 187–207 (2007).Google Scholar
[18]Liu, W. and Yan, N., A posteriori error estimates for distributed convex optimal control problems, Adv. Comput. Math. 15, 285–309 (2001).Google Scholar
[19]Liu, W. and Yan, N., A posteriori error estimates for optimal control problems governed by parabolic equations, Numer. Math. 93, 497–521 (2003).Google Scholar
[20]Liu, W. and Yan, N., Adaptive Finite Element Methods for Optimal Control Governed by PDEs, Science Press, Beijing (2008).Google Scholar
[21]Lu, Z., Chen, Y. and Zheng, W., A posteriori error estimates of lowest order Ravairt-Thomas mixed finite element methods for bilinear optimal control problems, East Asian J. Appl. Math. 2, 108–125 (2012).Google Scholar
[22]Meidner, D. and Vexler, B., A priori error estimates for space-time finite element discretisation of parabolic optimal control problems Part I: problems without control constraints, SIAM J. Control Optim. 47, 1150–1177 (2008).Google Scholar
[23]Meidner, D. and Vexler, B., A priori error estimates for space-time finite element discretisation of parabolic optimal control problems Part II: problems with control constraints, SIAM J. Control Optim. 47, 1301–1329 (2008).Google Scholar
[24]Meyer, C. and Rösch, A., Superconvergence properties of optimal control problems, SIAM J. Control Optim. 43 970–985 (2004).CrossRefGoogle Scholar
[25]Neittaanmaki, P. and Tiba, D., Optimal Control of Nonlinear Parabolic Systems: Theory, Algorithms and Applications, Dekker, New York (1994).Google Scholar
[26]Tang, Y. and Chen, Y., Superconvergence analysis of fully discrete finite element methods for semilinear parabolic optimal control problems, Front. Math. China 8 443–464, (2013).CrossRefGoogle Scholar
[27]Tang, Y. and Hua, Y., Superconvergence of fully discrete finite elements for parabolic control problems with integral constraints, East Asian J. Appl. Math. 3, 138–153 (2013).CrossRefGoogle Scholar
[28]Thomée, V., Galekin Finite Element Methods for Parabolic Problems, Springer Verlag, Berlin (1997).Google Scholar
[29]Tiba, D., Lectures on the Optimal Control of Elliptic Equations, University of Jyvaskyla Press, Finland (1995).Google Scholar
[30]Xing, X., Chen, Y. and Yi, N., Error estimates of mixed finite element methods for quadratic optimal control problems, J. Comput. Appl. Math. 233, 1812–1820 (2010).CrossRefGoogle Scholar
[31]Xiong, C. and Li, Y., A posteriori error estimates for optimal distributed control governed by the evolution equations, Appl. Numer. Math. 61, 181–200 (2011).Google Scholar
[32]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, 471–487 (2008).Google Scholar
[33]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, 2057–2066 (2010).Google Scholar
[34]Zienkiewicz, O.C. and Zhu, J.Z., The superconvergent patch recovery and a posteriori error estimates, Int. J. Numer. Methods Eng. 33, 1331–1382 (1992).Google Scholar
[35]Zienkiewicz, O.C. and Zhu, J.Z., The superconvergence patch recovery (SPR) and adaptive finite element refinement, Comput. Methods Appl. Mech. Eng. 101, 207–224 (1992).CrossRefGoogle Scholar