Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-21T20:43:53.507Z Has data issue: false hasContentIssue false
  • English
  • Français

Superconvergence of Fully Discrete Finite Elements for Parabolic Control Problems with Integral Constraints

Published online by Cambridge University Press:  28 May 2015

Y. Tang  and
Y. Hua
Y. Tang*
Affiliation:
Department of Mathematics and Computational Science, Hunan University of Science and Engineering, Yongzhou 425100, Hunan, China
Y. Hua*
Affiliation:
Department of Mathematics and Computational Science, Hunan University of Science and Engineering, Yongzhou 425100, Hunan, China
*
Corresponding author. Email: tangyuelonga@163.com
Corresponding author. Email: tangyuelonga@163.com
Get access

Abstract

A quadratic optimal control problem governed by parabolic equations with integral constraints is considered. A fully discrete finite element scheme is constructed for the optimal control problem, with finite elements for the spatial but the backward Euler method for the time discretisation. Some superconvergence results of the control, the state and the adjoint state are proved. Some numerical examples are performed to confirm theoretical results.

Keywords

35B3749J2065N30Superconvergencefinite element methodoptimal control problemsparabolic equationsintegral constraint
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 3 , Issue 2 , May 2013 , pp. 138 - 153
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]Chen, Y., Superconvergence of mixed finite element methods for optimal control problems, Math. Comp. 77 (2008) 1269–1291.CrossRefGoogle Scholar
[2]Chen, Y. and Dai, Y., Superconvergence for optimal control problems governed by semi-linear elliptic equations, J. Sci. Comput. 39 (2009) 206–221.Google Scholar
[3]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) 382–403.Google Scholar
[4]Chen, Y., Huang, Y. and Yi, N., A posteriori error estimates of spectral method for optimal control problems governed by parabolic equations, Sci. China Seri. A: Math. 51(8) (2008) 1376–1390.Google Scholar
[5]Fu, H. and Rui, H., A priori error estimates for optimal control problems governed by transient advection-diffusion equations, J. Sci. Comput. 38 (2009) 290–315.Google Scholar
[6]Hinze, M., Yan, N. and Zhou, Z., Variational discretisation for optimal control governed by convection dominated diffusion equations, J. Comput. Math. 27 (2009) 237–253.Google Scholar
[7]Hou, C., Chen, Y. and Lu, Z., Superconvergece property of finite element methods for parabolic optimal control problems, J. Ind. Manag. Optim. 7(4) (2011) 927–945.Google Scholar
[8]Li, B. and Zhang, Z., Analysis of a class of superconvergence patch recovery techniques for linear and bilinear finite elements, Numer. Meth. PDE. 15 (1999) 151–167.3.0.CO;2-O>CrossRefGoogle Scholar
[9]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) 155–182.CrossRefGoogle Scholar
[10]Lions, J., Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, Berlin, 1971.Google Scholar
[11]Lions, J. and Magenes, E., Non Homogeneous Boundary Value Problems and Applications, Springer-Verlag, Berlin, 1972.Google Scholar
[12]Liu, W. and Yan, N., A posteriori error estimates for optimal control problems governed by parabolic equations, Numer. Math. 93 (2003) 497–521.Google Scholar
[13]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(3) (2008) 1150–1177.Google Scholar
[14]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(3) (2008) 1301–1329.CrossRefGoogle Scholar
[15]Meyer, C. and Rösch, A., Superconvergence properties of optimal control problems, SIAM J. Control Optim. 43(3) (2004) 970–985.Google Scholar
[16]Tang, Y. and Chen, Y., Superconvergence analysis of fully discrete finite element methods for semilinear parabolic optimal control problems, Front. Math. China 8(2) (2013) 443–464.CrossRefGoogle Scholar
[17]Xiong, C. and Li, Y., A posteriori error estimates for optimal distributed control governed by the evolution equations, Appl. Numer. Math. 61 (2011) 181–200.CrossRefGoogle Scholar
[18]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) 471–487.Google Scholar
[19]Zhang, Z., Ultraconvergence of the patch recovery technique II, Math. Comp. 69 (2000) 141–158.Google Scholar
[20]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) 2057–2066.Google Scholar
[21]Zienkiwicz, O.C. and Zhu, J.Z., The superconvergence patch recovery and a posteriori error estimates, Int. J. Numer. Meth. Eng. 33 (1992) 1331–1382.Google Scholar
[22]Zienkiwicz, O.C. and Zhu, J.Z., The superconvergence patch recovery (SPR) and adaptive finite element refinement, Comput. Meth. Appl. Math. 101 (1992) 207–224.Google Scholar