# American Institute of Mathematical Sciences

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

 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 $h^{2}$.
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.

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##### 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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