Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-27T23:51:59.908Z Has data issue: false hasContentIssue false
  • English
  • Français

A Posteriori Error Estimates of Semidiscrete Mixed Finite Element Methods for Parabolic Optimal Control Problems

Published online by Cambridge University Press:  06 March 2015

Yanping Chen  and
Zhuoqing Lin
Yanping Chen*
Affiliation:
School of Mathematical Sciences, South China Normal University, Guangzhou 510631, Guangdong, P.R. China
Zhuoqing Lin
Affiliation:
School of Mathematical Sciences, South China Normal University, Guangzhou 510631, Guangdong, P.R. China
*
*Corresponding author. Email addresses: yanpingchen@scnu.edu.cn (Y. Chen), 0626lzq@sina.com (Z. Lin)
Get access

Abstract

A posteriori error estimates of semidiscrete mixed finite element methods for quadratic optimal control problems involving linear parabolic equations are developed. The state and co-state are discretised by Raviart-Thomas mixed finite element spaces of order k, and the control is approximated by piecewise polynomials of order k (k ≥ 0). We derive our a posteriori error estimates for the state and the control approximations via a mixed elliptic reconstruction method. These estimates seem to be unavailable elsewhere in the literature, although they represent an important step towards developing reliable adaptive mixed finite element approximation schemes for the control problem.

Keywords

49J2065N30A posteriori error estimatesoptimal control problemsparabolic equationselliptic reconstructionsemidiscrete mixed finite element methods
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 5 , Issue 1 , February 2015 , pp. 85 - 108
Copyright
Copyright © Global-Science Press 2015 

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]Arada, N., Casas, E. and Tröltzsch, F.A., Error estimates for the numerical approximation of a semilinear elliptic control problem, Comp. Optim. Appl. 23, 201–229 (2002)CrossRefGoogle Scholar
[2]Babuska, I. and Strouboulis, T., The Finite Element Method and its Reliability, Oxford University Press (2001)CrossRefGoogle Scholar
[3]Babuska, I. and Ohnimus, S., A posteriori error estimation for the semidiscrete finite element method of parabolic partial differential equations, Comp. Methods Appl. Mech. Eng. 190, 65–187 (2001)CrossRefGoogle Scholar
[4]Babuska, I., Feistauer, M. and Solin, P., On one approach to a posteriori error estimates for evolution problems solved by the method of lines, Num. Math. 89, 225–256 (2001)Google Scholar
[5]Brunner, H. and Yan, N., Finite element methods for optimal control problems governed by integral equations and integro-differential equations, Num. Math. 101, 1–27 (2005)CrossRefGoogle Scholar
[6]Brezzi, F. and Fortin, M., Mixed and Hybrid Finite Element Methods, Springer, New York (1991)Google Scholar
[7]Chen, Y., Superconvergence of quadratic optimal control problems by triangular mixed finite elements, Int. J. Num. Meths. Eng. 75, 881–898 (2008)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. Sc. Comp. 42, 382–403 (2009)CrossRefGoogle Scholar
[9]Chen, Y., Yi, N. and Liu, W.B., A Legendre Galerkin spectral method for optimal control problems governed by elliptic equations, SIAM J. Num. Anal. 46, 2254–2275 (2008)CrossRefGoogle Scholar
[10]Chen, Y. and Liu, W.B., A posteriori error estimates for mixed finite element solutions of convex optimal control problems, J. Comp. Appl. Math. 211, 76–89 (2008)Google Scholar
[11]Carstensen, C., A posteriori error estimate for the mixed finite element method, Math. Comp. 66, 465–476 (1997)CrossRefGoogle Scholar
[12]Douglas, J. and Roberts, J.E., Global estimates for mixed finite element methods for second order elliptic equations, Math. Comp. 44, 39–52 (1985)Google Scholar
[13]Demlow, A., Lakkis, O. and Makridakes, C., A posteriori eror estimates in maximum norm for parabolic problems, SIAM J. Num. Anal. 47, 2157–2176 (2009)Google Scholar
[14]Eriksson, K. and Johnson, C., Adaptive finite elements methods for parabolic problems I: A linear model problem, SIAM J. Num. Anal. 28, 43–77 (1991)CrossRefGoogle Scholar
[15]Eriksson, K. and Johnson, C., Adaptive finite element methods for parabolic problems IV: Nonlinear problems, SIAM J. Num. Anal. 32, 1729–1749 (1995)Google Scholar
[16]Georgoulis, E., Lakkis, O. and Virtanen, J.M., A posteriori error control for discontinuous Galerkin methods for parabolic problems, SIAM J. Num. Anal. 49, 427–458 (2011)Google Scholar
[17]Gong, W. and Yan, N., A posteriori error estimate for boundary control problems governed by the parabolic partial differential equations, J. Comp. Math. 27, 68–88 (2009)Google Scholar
[18]Hou, L. and Turner, J.C., Analysis and finite element approximation of an optimal control problem in electrochemistry with current density controls, Num. Math. 71, 289–315 (1995)Google Scholar
[19]Haslinger, J. and Neittaanmaki, P., Finite Element Approximation for Optimal Shape Design, John Wiley and Sons, Chichester (1989)Google Scholar
[20]Hoppe, R.H.W., Iliash, Y., Iyyunni, C. and Sweilam, N.H., A posteriori error estimates for adaptive finite element discretizations of boundary control problems, J. Num. Math. 14, 57–82 (2006)Google Scholar
[21]Johnson, C., Nie, Y. and Thomée, V., An a posteriori error estimate and adaptive time step control for a backward Euler discretisation of a parabolic problem, SIAM J. Num. Anal. 27, 277–291 (1990)CrossRefGoogle Scholar
[22]Knowles, G., Finite element approximation of parabolic time optimal control problems, SIAM J. Control Optim. 20, 414–427 (1982)Google Scholar
[23]Lions, J., Optimal Control of Systems Governed by Partial Differential Equations, Springer, Berlin (1971)Google Scholar
[24]Lions, J. and Magenes, E., Non-homogeneous Boundary Value Problems and Applications, Grandlehre B. 181, Springer (1972)Google Scholar
[25]Liu, W.B., Ma, H., Tang, T. and Yan, N., A posteriori error estimates for discontinuous Galerkin time-stepping method for optimal control problems governed by parabolic equations, SIAM J. Num. Anal. 42, 1032–1061 (2004)Google Scholar
[26]Liu, W.B. and Yan, N., A posteriori error estimates for convex boundary control problems, SIAM J. Num. Anal. 39, 73–99 (2001)Google Scholar
[27]Liu, W.B. and Yan, N., A posteriori error analysis for convex distributed optimal control problems, Adv. Comp. Math. 15, 285–309 (2001)Google Scholar
[28]Liu, W.B. and Yan, N., A posteriori error estimates for optimal control problems governed by Stokes equations, SIAM J. Num. Anal. 40, 1850–1869 (2003)Google Scholar
[29]Liu, W.B. and Yan, N., A posteriori error estimates for optimal control problems governed by parabolic equations, Num. Math. 93, 497–521 (2003)Google Scholar
[30]Li, R., Liu, W.B., Ma, H. and Tang, T., Adaptive finite element approximation of elliptic control problems, SIAM J. Control Optim. 41, 1321–1349 (2002)Google Scholar
[31]Lakkis, O. and Makridakis, C., Elliptic reconstruction and a posteriori error estimates for fully discrete linear parabolic problems, Math. Comp. 75, 1627–1658 (2006)Google Scholar
[32]Mcknight, R. and Bosarge, W. Jr., The Ritz-Galerkin procedure for parabolic control problems, SIAM J. Control Optim. 11, 510–524 (1973)Google Scholar
[33]Makridakis, C. and Nochetto, R.H., Elliptic reconstruction and a posteriori error estimates for parabolic problems, SIAM J. Num. Anal. 41, 1585–1594 (2003)Google Scholar
[34]Nochetto, R.H., Savaré, G. and erdi, C. V, A posteriori error estimates for variable time step discretizations of nonlinear evolution equations, Comm. Pure Appl. Math. 53, 525–589 (2000)3.0.CO;2-M>CrossRefGoogle Scholar
[35]Neittaanmaki, P. and Tiba, D., Optimal Control of Nonlinear Parabolic Systems: Theory, Algorithms and Applications, Dekker, New York (1994)Google Scholar
[36]Tiba, D., Lectures on the Optimal Control of Elliptic Problems, University of Jyvaskyla Press, Finland (1995)Google Scholar
[37]Tröltzsch, F., Semidiscrete Ritz-Galerkin approximation of nonlinear parabolic boundary control problems-strong convergence of optimal control, Appl. Math. Optim. 29, 309–329 (1994)Google Scholar
[38]Verfürth, R., A posteriori error estimates for nonlinear problems: Lr(0, T; Lρ(Ω))-error estimates for finite element discretisation of parabolic equations, Num. Meth. PDE. 14, 487–518 (1998)Google Scholar
[39]Verfürth, R., A posteriori error estimates for nonlinear problems: Lr(0, T; Lρ(Ω))-error estimates for finite element discretisation of parabolic equations, Math. Comp. 67, 1335–1360 (1998)Google Scholar
[40]Thomée, V., Galerkin Finite Element Methods for Parabolic Problems, Springer (1997)Google Scholar
[41]Wheeler, M.F., A priori L2 error estimates for Galerkin approximations to parabolic partial differential equations, SIAM J. Num. Anal. 10, 723–759 (1973)Google Scholar