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July  2013, 9(3): 631-642. doi: 10.3934/jimo.2013.9.631

Superconvergence for elliptic optimal control problems discretized by RT1 mixed finite elements and linear discontinuous elements

Tianliang Hou 1, and  Yanping Chen 2,

1. 

School of Mathematical Sciences, South China Normal University, Guangzhou 510631, Guangdong, China
2. 

School of Mathematical Sciences, South China Normal University, Guangzhou 510631

Received  April 2012 Revised  March 2013 Published  April 2013

In this paper, we investigate the superconvergence property of a quadratic elliptic control problem with pointwise control constraints. The state and the co-state variables are approximated by the Raviart-Thomas mixed finite element of order $k=1$ and the control variable is discretized by piecewise linear but discontinuous functions. Approximations of the optimal solution of the continuous optimal control problem will be constructed by a projection of the discrete adjoint state. It is proved that these approximations have convergence order $h^{2}$.
Keywords: postprocessing., discontinuous functions, mixed finite element methods, Elliptic equations, optimal control problems, superconvergence.
Mathematics Subject Classification: 49J20, 65N3.
Citation: Tianliang Hou, Yanping Chen. Superconvergence for elliptic optimal control problems discretized by RT1 mixed finite elements and linear discontinuous elements. Journal of Industrial and Management Optimization, 2013, 9 (3) : 631-642. doi: 10.3934/jimo.2013.9.631
References:
[1]

N. Arada, E. Casas and F. Tröltzsch, Error estimates for the numerical approximation of a semilinear elliptic control problem, Comput. Optim. Appl., 23 (2002), 201-229. doi: 10.1023/A:1020576801966.

[2]

R. Becker, H. Kapp and R. Rannacher, Adaptive finite element methods for optimal control of partial differential equations: Basic concept, SIAM J. Control Optim., 39 (2000), 113-132. doi: 10.1137/S0363012999351097.

[3]

F. Brezzi and M. Fortin, "Mixed and Hybrid Finite Element Methods," Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-3172-1.

[4]

E. Casas and F. Tröltzsch, Error estimates for linear-quadratic elliptic control problems, in "Analysis and Optimization of Differential Systems," Kluwer Academic Publishers, (2003), 89-100.

[5]

Y. Chen, Superconvergence of mixed finite element methods for optimal control problems, Math. Comp., 77 (2008), 1269-1291. doi: 10.1090/S0025-5718-08-02104-2.

[6]

Y. Chen, Superconvergence of quadratic optimal control problems by triangular mixed finite elements, Inter. J. Numer. Meths. Eng., 75 (2008), 881-898. doi: 10.1002/nme.2272.

[7]

Y. Chen, Y. Huang, W. B. Liu and N. Yan, Error estimates and superconvergence of mixed finite element methods for convex optimal control problems, J. Sci. Comput., 42 (2009), 382-403. doi: 10.1007/s10915-009-9327-8.

[8]

Y. Chen and Y. Q. Dai, Superconvergence for optimal control problems governed by semi-linear elliptic equations, J. Sci. Comput., 39 (2009), 206-221. doi: 10.1007/s10915-008-9258-9.

[9]

P. G. Ciarlet, "The Finite Element Method for Elliptic Problems," North-Holland, Amsterdam, 1978.

[10]

P. Grisvard, "Elliptic Problems in Nonsmooth Domains," Pitman, Boston-London-Melbourne, 1985.

[11]

J. Douglas and J. E. Roberts, Global estimates for mixed finite element methods for second order elliptic equations, Math. Comp., 44 (1985), 39-52. doi: 10.2307/2007791.

[12]

F. S. Falk, Approximation of a class of optimal control problems with order of convergence estimates, J. Math. Anal. Appl., 44 (1973), 28-47.

[13]

T. Geveci, On the approximation of the solution of an optimal control problem governed by an elliptic equation, RAIRO. Anal. Numer., 13 (1979), 313-328.

[14]

R. Li, W. B. Liu, H. P. Ma and T. Tang, Adaptive finite element approximation of elliptic optimal control, SIAM J. Control Optim., 41 (2002), 1321-1349. doi: 10.1137/S0363012901389342.

[15]

R. Li and W., Liu, http://dsec.pku.edu.cn/~rli/.

[16]

J. L. Lions, "Optimal Control of Systems Governed by Partial Differential Equations," Springer-Verlag, Berlin, 1971.

[17]

W. B. Liu and N. N. Yan, A posteriori error analysis for convex distributed optimal control problems, Adv. Comp. Math., 15 (2001), 285-309. doi: 10.1023/A:1014239012739.

[18]

C. Meyer and A. Rösch, Superconvergence properties of optimal control problems, SIAM J. Control Optim., 43 (2004), 970-985. doi: 10.1137/S0363012903431608.

[19]

P. A. Raviart and J. M. Thomas, A mixed finite element method for 2nd order elliptic problems, in "Aspects of the Finite Element Method" Springer, (1977), 292-315.

[20]

A. Rösch and R. Simon, Linear and discontinuous approximations for optimal control problems, Numer. Funct. Anal. Optim., 26 (2005), 427-448. doi: 10.1081/NFA-200067309.

show all references

References:
[1]

N. Arada, E. Casas and F. Tröltzsch, Error estimates for the numerical approximation of a semilinear elliptic control problem, Comput. Optim. Appl., 23 (2002), 201-229. doi: 10.1023/A:1020576801966.

[2]

R. Becker, H. Kapp and R. Rannacher, Adaptive finite element methods for optimal control of partial differential equations: Basic concept, SIAM J. Control Optim., 39 (2000), 113-132. doi: 10.1137/S0363012999351097.

[3]

F. Brezzi and M. Fortin, "Mixed and Hybrid Finite Element Methods," Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-3172-1.

[4]

E. Casas and F. Tröltzsch, Error estimates for linear-quadratic elliptic control problems, in "Analysis and Optimization of Differential Systems," Kluwer Academic Publishers, (2003), 89-100.

[5]

Y. Chen, Superconvergence of mixed finite element methods for optimal control problems, Math. Comp., 77 (2008), 1269-1291. doi: 10.1090/S0025-5718-08-02104-2.

[6]

Y. Chen, Superconvergence of quadratic optimal control problems by triangular mixed finite elements, Inter. J. Numer. Meths. Eng., 75 (2008), 881-898. doi: 10.1002/nme.2272.

[7]

Y. Chen, Y. Huang, W. B. Liu and N. Yan, Error estimates and superconvergence of mixed finite element methods for convex optimal control problems, J. Sci. Comput., 42 (2009), 382-403. doi: 10.1007/s10915-009-9327-8.

[8]

Y. Chen and Y. Q. Dai, Superconvergence for optimal control problems governed by semi-linear elliptic equations, J. Sci. Comput., 39 (2009), 206-221. doi: 10.1007/s10915-008-9258-9.

[9]

P. G. Ciarlet, "The Finite Element Method for Elliptic Problems," North-Holland, Amsterdam, 1978.

[10]

P. Grisvard, "Elliptic Problems in Nonsmooth Domains," Pitman, Boston-London-Melbourne, 1985.

[11]

J. Douglas and J. E. Roberts, Global estimates for mixed finite element methods for second order elliptic equations, Math. Comp., 44 (1985), 39-52. doi: 10.2307/2007791.

[12]

F. S. Falk, Approximation of a class of optimal control problems with order of convergence estimates, J. Math. Anal. Appl., 44 (1973), 28-47.

[13]

T. Geveci, On the approximation of the solution of an optimal control problem governed by an elliptic equation, RAIRO. Anal. Numer., 13 (1979), 313-328.

[14]

R. Li, W. B. Liu, H. P. Ma and T. Tang, Adaptive finite element approximation of elliptic optimal control, SIAM J. Control Optim., 41 (2002), 1321-1349. doi: 10.1137/S0363012901389342.

[15]

R. Li and W., Liu, http://dsec.pku.edu.cn/~rli/.

[16]

J. L. Lions, "Optimal Control of Systems Governed by Partial Differential Equations," Springer-Verlag, Berlin, 1971.

[17]

W. B. Liu and N. N. Yan, A posteriori error analysis for convex distributed optimal control problems, Adv. Comp. Math., 15 (2001), 285-309. doi: 10.1023/A:1014239012739.

[18]

C. Meyer and A. Rösch, Superconvergence properties of optimal control problems, SIAM J. Control Optim., 43 (2004), 970-985. doi: 10.1137/S0363012903431608.

[19]

P. A. Raviart and J. M. Thomas, A mixed finite element method for 2nd order elliptic problems, in "Aspects of the Finite Element Method" Springer, (1977), 292-315.

[20]

A. Rösch and R. Simon, Linear and discontinuous approximations for optimal control problems, Numer. Funct. Anal. Optim., 26 (2005), 427-448. doi: 10.1081/NFA-200067309.

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