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Superconvergence and L∞-Error Estimates of RT1 Mixed Methods for Semilinear Elliptic Control Problems with an Integral Constraint

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:
Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Department of Mathematics, Xiangtan University, Xiangtan 411105, Hunan, China
*
Corresponding author.Email address:yanpingchen@scnu.edu.cn
Corresponding author.Email address:htlchb@163.com
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Abstract

In this paper, we investigate the superconvergence property and the L∞-error estimates of mixed finite element methods for a semilinear elliptic control problem with an integral constraint. The state and co-state are approximated by the order one Raviart-Thomas mixed finite element space and the control variable is approximated by piecewise constant functions or piecewise linear functions. We derive some super-convergence results for the control variable and the state variables when the control is approximated by piecewise constant functions. Moreover, we derive L∞-error estimates for both the control variable and the state variables when the control is discretized by piecewise linear functions. Finally, some numerical examples are given to demonstrate the theoretical results.

Keywords

49J2065N30Semilinear elliptic equationsoptimal control problemssuperconvergenceL∞-error estimatesmixed finite element methodspostprocessing
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 5 , Issue 3 , August 2012 , pp. 423 - 446
Copyright
Copyright © Global Science Press Limited 2012

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References

[1]Arada, N., Casas, E. and Tröltzsch, F., Error estimates for the numerical approximation of a semilinear elliptic control problem, Comp. Optim. Appl., 23 (2002), pp. 201–229.CrossRefGoogle Scholar
[2]Brezzi, F. and Fortin, M., Mixed and hybrid finite element methods, Springer-Verlag., 95 (1991), pp. 65–187.Google Scholar
[3]Chen, Y., Superconvergence of mixed finite element methods for optimal control problems, Math. Comp., 77 (2008), pp. 1269–1291.CrossRefGoogle 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.CrossRefGoogle Scholar
[5]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.Google 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., 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
[8]Ciarlet, P. G., The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978.Google Scholar
[9]Deng, K., Chen, Y. and Lu, Z., Higher order triangular mixed finite element methods for semilinear quadratic optimal control problems, Numer. Math. Theor. Meth. Appl., 4 (2) (2011), pp. 180–196.CrossRefGoogle Scholar
[10]Douglas, J. and Roberts, J. E., Global estimates for mixed finite element methods for second order elliptic equations, Math. Comp., 44 (1985), pp. 39–52.CrossRefGoogle Scholar
[11]Falk, F. S., Approximation of a class of optimal control problems with order of convergence estimates, J. Math. Anal. Appl., 44 (1973), pp. 28–47.CrossRefGoogle Scholar
[12]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
[13]Geveci, T., On the approximation of the solution of an optimal control problem governed by an elliptic equation, RAIRO. Anal. Numer., 13 (1979), pp. 313–328.Google Scholar
[14]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
[15]Knowles, G., Finite element approximation of parabolic time optimal control problems, SIAM J. Control Optim., 20 (1982), pp. 414–427.CrossRefGoogle Scholar
[16]Li, R. and Liu, W., http://circus.math.pku.edu.cn/AFEPack.Google Scholar
[17]Lions, J. L., Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, Berlin, 1971.CrossRefGoogle Scholar
[18]Ladyzhenskaya, O. A. and Uraltseva, N., Linear and Quasilinear Elliptic Equations, Academic Press, New York, 1968.Google Scholar
[19]Lu, Z. and Chen, Y., L∞-error estimates of triangular mixed finite element methods for optimal control problems governed by semilinear elliptic equations, Numer. Anal. Appl., 12 (1) (2009), pp. 74–86.Google Scholar
[20]Meyer, C. and Rösch, A., Superconvergence properties of optimal control problems, SIAM J. Control Optim., 43 (3) (2004), pp. 970–985.CrossRefGoogle Scholar
[21]Meyer, C. and A. RöSCH, L∞-error estimates for approximated optimal control problems, SIAM J. Control Optim., 44 (2005), pp. 1636–1649.CrossRefGoogle Scholar
[22]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
[23]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
[24]Mckinght, R. S. and Borsarge, J., The Rite-Galerkin procedure for parabolic control problems, SIAM J. Control Optim., 11 (1973), pp. 510–542.Google Scholar
[25]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