Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-22T18:53:59.939Z Has data issue: false hasContentIssue false
  • English
  • Français

Error Estimates and Superconvergence of RT0 Mixed Methods for a Class of Semilinear Elliptic Optimal Control Problems

Published online by Cambridge University Press:  28 May 2015

Yanping Chen  and
Tianliang Hou
Yanping Chen*
Affiliation:
School of Mathematical Sciences, South China Normal University, Guangzhou 510631, Guangdong, China
Tianliang Hou*
Affiliation:
School of Mathematical Sciences, South China Normal University, Guangzhou 510631, Guangdong, China Research Institute of Hong Kong Baptist University in Shenzhen, Shenzhen 518057, Guangdong, China
*
Corresponding author.Email address:[email protected]
Corresponding author.Email address:[email protected]
Get access

Abstract

In this paper, we will investigate the error estimates and the superconvergence property of mixed finite element methods for a semilinear elliptic control problem with an integral constraint on control. The state and co-state are approximated by the lowest order Raviart-Thomas mixed finite element and the control variable is approximated by piecewise constant functions. We derive some superconvergence properties for the control variable and the state variables. Moreover, we derive L∞- 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

49J2065N30Semilinear elliptic equationsoptimal control problemssuperconvergenceerror estimatesmixed finite element methods
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 6 , Issue 4 , November 2013 , pp. 637 - 656
Copyright
Copyright © Global Science Press Limited 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]Brezzi, F. and Fortin, M., Mixed and Hybrid Finite Element Methods, Springer-Verlag., 95 (1991), pp. 65–187.CrossRefGoogle Scholar
[2]Bonnans, J. F., Second-order analysis for control constrained optimal control problems of semi-linear elliptic systems, Appp. Math. Optim., 38 (1998), pp. 303–325.CrossRefGoogle Scholar
[3]Bonnans, J. F. and Casas, E., An extension of p ontry agin’s principle for state constrained optimal control of semilinear elliptic equation and variational inequalities, SIAM J. Control Optim., 33 (1995), pp. 274–298.CrossRefGoogle Scholar
[4]Chen, Y., Superconvergence of mixed finite element methods for optimal control problems, Math. Comput., 77 (2008), pp. 1269–1291.CrossRefGoogle Scholar
[5]Chen, Y., Superconvergence of quadratic optimal control problems by triangular mixed finite elements, Inter. J. Numer. Meths. Eng., 75 (8) (2008), pp, 881–898.CrossRefGoogle Scholar
[6]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
[7]Chen, Y., Huang, F., Yi, N. and Liu, W. B., A Legendre-Galerkin spectral method for optimal control problems governed by Stokes equations, SIAM J. Numer. Anal., 49 (2011), pp. 1625–1648.CrossRefGoogle Scholar
[8]Chen, Y. and Hou, T., Superconvergence and L∞-error estimates of RT1 mixed methods for semi-linear elliptic control problems with an integral constraint, Numer. Math. Theor. Meth. Appl., 5 (2012), pp. 423–446.Google Scholar
[9]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. Sci. Comput., 42 (3) (2009), pp. 382–403.CrossRefGoogle Scholar
[10]Chen, Y., Yi, N. and Liu, W. B., A Legendre Galerkin spectral method for optimal control problems governed by elliptic equations, SIAM J. Numer. Anal., 46 (2008), pp. 2254–2275.CrossRefGoogle Scholar
[11]Ciarlet, P. G., The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978.Google Scholar
[12]Deng, K., Chen, Y and Lu, Z., Higher order triangular mixedfinite element methods for semilinear quadratic optimal control problems, Numer. Math. Theor. Meth. Appl., 4(2) (2011), pp. 180–196.CrossRefGoogle Scholar
[13]Douglas, J. and Roberts, J. E., Global estimates for mixedfinite element methods for second order elliptic equations, Math. Comput., 44 (1985), pp. 39–52.CrossRefGoogle Scholar
[14]Ewing, R. E., Liu, M. M. and Wang, J., Superconvergence of mixedfinite element approximations over quadrilaterals, SIAM J. Numer. Anal., 36 (1999), pp. 772–787.CrossRefGoogle Scholar
[15]Li, R. and Liu, W., http://circus.math.pku.edu.cn/AFEPack.Google Scholar
[16]Lin, Q. and Yan, N., Structure and Analysis for Efficient Finite Element Methods, Publishers of Hebei University, China, 1996.Google Scholar
[17]Li, R., Liu, W. B. 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
[18]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
[19]Lions, J. L., Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, Berlin, 1971.CrossRefGoogle Scholar
[20]Lu, Z. and Chen, Y, L∞-error estimates of triangular mixedfinite element methods for optimal control problems governed by semilinear elliptic equations, Numer. Anal. Appl., 12(1) (2009), pp. 74-86.Google Scholar
[21]Meyer, C. and Rösch, A., Superconvergence properties of optimal control problems, SIAM J. Control Optim., 43(3) (2004), pp. 970-985.CrossRefGoogle Scholar
[22]Raviart, P. A. and Thomas, J. M., A mixed finite element method for 2nd order elliptic problems, aspects of the finite element method, Lecture Notes in Math, Springer, Berlin, 606 (1977), pp. 292–315.Google Scholar
[23]Yang, D., Chang, Y. and Liu, W. B., A priori error estimates and superconvergence analysis for an optimal control problem of bilinear type, J. Comput. Math., 4 (2008), pp. 471–487.Google Scholar
[24]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