Hostname: page-component-8448b6f56d-qsmjn Total loading time: 0 Render date: 2024-04-18T20:44:28.178Z Has data issue: false hasContentIssue false
  • English
  • Français

Error Estimates of Mixed Methods for Optimal Control Problems Governed by General Elliptic Equations

Part of: Partial differential equations, boundary value problems Existence theories

Published online by Cambridge University Press:  19 September 2016

Tianliang Hou  and
Li Li
Tianliang Hou*
Affiliation:
School of Mathematics and Statistics, Beihua University, Jilin 132013, China
Li Li*
Affiliation:
Key Laboratory for Nonlinear Science and System Structure, School of Mathematics and Statistics, Chongqing Three Gorges University, Wanzhou 404100, China
*
*Corresponding author. Email:htlchb@163.com (T. L. Hou), zyxlily81@126.com (L. Li)
*Corresponding author. Email:htlchb@163.com (T. L. Hou), zyxlily81@126.com (L. Li)
Get access

Abstract

In this paper, we investigate the error estimates of mixed finite element methods for optimal control problems governed by general elliptic equations. The state and co-state are approximated by the lowest order Raviart-Thomas mixed finite element spaces and the control variable is approximated by piecewise constant functions. We derive L2 and H–1-error estimates both for the control variable and the state variables. Finally, a numerical example is given to demonstrate the theoretical results.

Keywords

General elliptic equationsoptimal control problemssuperconvergenceerror estimatesmixed finite element methods

MSC classification

Secondary: 49J20: Optimal control problems involving partial differential equations 65N30: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 8 , Issue 6 , December 2016 , pp. 1050 - 1071
Copyright
Copyright © Global-Science Press 2016 

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] Bonnans, J. F. and Casas, E., An extension of Pontryagin's principle for state constrained optimal control of semilinear elliptic eqnation and variational inequalities, SIAM J. Control Optim., 33 (1995), pp. 274–298.Google Scholar
[2] Brezzi, F. and Fortin, M., Mixed and Hybrid Finite Element Methods, Springer-Verlag, New York, 1991.Google Scholar
[3] Chen, Y., Superconvergence of mixed finite element methods for optimal control problems, Math. Comput., 77 (2008), pp. 1269–1291.Google Scholar
[4] Chen, Y., Superconvergence of quadratic optimal control problems by triangular mixed finite elements, Inter. J. Numer. Meths. Eng., 75(8) (2008), pp. 881–898.Google Scholar
[5] Chen, Y. and Dai, Y., Superconvergence for optimal control problems governed by semi-linear elliptic equations, J. Sci. Comput., 39 (2009), pp. 206–221.Google Scholar
[6] 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(3) (2010), pp. 382–403.Google Scholar
[7] Ciarlet, P. G., The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978.Google Scholar
[8] Douglas, J. and Roberts, J. E., Global estimates for mixed finite element methods for second order elliptic equations, Math. Comput., 44 (1985), pp. 39–52.Google Scholar
[9] 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.CrossRefGoogle Scholar
[10] Hou, L. and Turner, J. C., Analysis and finite element approximation of an optimal control problem in electrochemistry with current density controls, Numer. Math., 71 (1995), pp. 289–315.Google Scholar
[11] Knowles, G., Finite element approximation of parabolic time optimal control problems, SIAM J. Control Optim., 20 (1982), pp. 414–427.Google Scholar
[12] Li, R. and Liu, W., http://circus.math.pku.edu.cn/AFEPack.Google Scholar
[13] Li, R., Liu, W., Ma, H. and Tang, T., Adaptive finite element approximation of elliptic control problems, SIAM J. Control Optim., 41 (2002), pp. 1321–1349.Google Scholar
[14] Lions, J. L., Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, Berlin, 1971.CrossRefGoogle Scholar
[15] Meyer, C. and Rösch, A., Superconvergence properties of optimal control problems, SIAM J. Control Optim., 43(3) (2004), pp. 970–985.Google Scholar
[16] Meyer, C. and Rösch, A., L∞-error estimates for approximated optimal control problems, SIAM J. Control Optim., 44 (2005), pp. 1636–1649.Google Scholar
[17] Meider, 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 (2008), pp. 1150–1177.Google Scholar
[18] Meider, 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 (2008), pp. 1301–1329.Google Scholar
[19] McKinght, R. S. and Borsarge, J., The Ritz-Galerkin procedure for parabolic control problems, SIAM J. Control Optim., 11 (1973), pp. 510–542.CrossRefGoogle Scholar
[20] Raviart, P. A. and Thomas, J. M., A mixed finite element method for 2nd order elliptic problems, Aspecs of the Finite Element Method, Lecture Notes in Math, Springer, Berlin, 606 (1977), pp. 292–315.Google Scholar
[21] Yang, D., Chang, Y. and Liu, W., A priori error estimates and superconvergence analysis for an optimal control problems of bilinear type, J. Comput. Math., 4 (2008), pp. 471–487.Google Scholar
[22] Yan, N., Superconvergence analysis and a posteriori error estimation of a finite element method for an optimal control problem governed by integral equations, Appl. Math., 54 (2009), pp. 267–283.CrossRefGoogle Scholar