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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 & 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. doi: 10.1023/A:1020576801966. Google Scholar

[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. doi: 10.1137/S0363012999351097. Google Scholar

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

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E. Casas and F. Tröltzsch, Error estimates for linear-quadratic elliptic control problems,, in, (2003), 89. Google Scholar

[5]

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

[6]

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

[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. doi: 10.1007/s10915-009-9327-8. Google Scholar

[8]

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

[9]

P. G. Ciarlet, "The Finite Element Method for Elliptic Problems,", North-Holland, (1978). Google Scholar

[10]

P. Grisvard, "Elliptic Problems in Nonsmooth Domains,", Pitman, (1985). Google Scholar

[11]

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

[12]

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

[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. Google Scholar

[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. doi: 10.1137/S0363012901389342. Google Scholar

[15]

R. Li and W., Liu,, , (). Google Scholar

[16]

J. L. Lions, "Optimal Control of Systems Governed by Partial Differential Equations,", Springer-Verlag, (1971). Google Scholar

[17]

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

[18]

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

[19]

P. A. Raviart and J. M. Thomas, A mixed finite element method for 2nd order elliptic problems,, in, (1977), 292. Google Scholar

[20]

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

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. doi: 10.1023/A:1020576801966. Google Scholar

[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. doi: 10.1137/S0363012999351097. Google Scholar

[3]

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

[4]

E. Casas and F. Tröltzsch, Error estimates for linear-quadratic elliptic control problems,, in, (2003), 89. Google Scholar

[5]

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

[6]

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

[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. doi: 10.1007/s10915-009-9327-8. Google Scholar

[8]

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

[9]

P. G. Ciarlet, "The Finite Element Method for Elliptic Problems,", North-Holland, (1978). Google Scholar

[10]

P. Grisvard, "Elliptic Problems in Nonsmooth Domains,", Pitman, (1985). Google Scholar

[11]

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

[12]

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

[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. Google Scholar

[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. doi: 10.1137/S0363012901389342. Google Scholar

[15]

R. Li and W., Liu,, , (). Google Scholar

[16]

J. L. Lions, "Optimal Control of Systems Governed by Partial Differential Equations,", Springer-Verlag, (1971). Google Scholar

[17]

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

[18]

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

[19]

P. A. Raviart and J. M. Thomas, A mixed finite element method for 2nd order elliptic problems,, in, (1977), 292. Google Scholar

[20]

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

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