October  2011, 7(4): 927-945. doi: 10.3934/jimo.2011.7.927

Superconvergence property of finite element methods for parabolic optimal control problems

1. 

Huashang College Guangdong University Of Business Studies, Guangzhou 511300, China

2. 

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

3. 

School of Mathematics and Statistics, Chongqing Three Gorges University, Chongqing 404000, China

Received  October 2010 Revised  June 2011 Published  August 2011

In this paper, a finite element method for a parabolic optimal control problem is introduced and analyzed. For the discretization of a quadratic convex optimal control problem, the state and co-state are approximated by piecewise linear functions and the control is approximated by piecewise constant functions. As a result, it is proved in this paper that the difference between a suitable interpolation of the control and its finite element approximation has superconvergence property in order $O(h^2)$. Finally, two numerical examples are presented to confirm our theoretical results.
Citation: Chunjuan Hou, Yanping Chen, Zuliang Lu. Superconvergence property of finite element methods for parabolic optimal control problems. Journal of Industrial & Management Optimization, 2011, 7 (4) : 927-945. doi: 10.3934/jimo.2011.7.927
References:
[1]

A. B. Andreev and R. D. Lazarov, Superconvergence of the gradient for quadratic triangular finite elements,, Numer. Methods PDEs, 4 (1988), 15.

[2]

J. H. Bramble and A. H. Schatz, Higher order local accuracy by averaging in the finite element method,, Math. Comp., 31 (1977), 94.

[3]

C. M. Chen and Y. Q. Huang, "High Accuracy Theory of Finite Element Methods,", Hunan Science and Technology Press, (1995).

[4]

C. M. Chen and V. Thomée, The lumped mass finite element method for a parabolic problem,, J. Austral. Math. Soc. Ser. B, 26 (1985), 329. doi: 10.1017/S0334270000004549.

[5]

Y. Chen, Superconvergence of quadratic optimal control problems by triangular mixed finite element methods,, Int. J. Numer. Methods Engineering, 75 (2008), 881. doi: 10.1002/nme.2272.

[6]

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.

[7]

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

[8]

Y. Chen and W. B. Liu, Error estimates and superconvergence of mixed finite element for quadratic optimal control,, Int. J. Numer. Anal. Model., 3 (2006), 311.

[9]

Y. Chen, N. Y. Yi and W. B. Liu, A Legendre-Galerkin spectral method for optimal control problems governed by elliptic equations,, SIAM J. Numer. Anal., 46 (2008), 2254. doi: 10.1137/070679703.

[10]

P. G. Ciarlet, "The Finite Element Method for Elliptic Problems,", Studies in Mathematics and its Applications, 4 (1978).

[11]

J. Jr. Douglas and T. Dupont, Some superconvergence results for Galerkin methods for the approximate solution of two-point boundary problems,, Topics in Numerical Analysis (Proc. Roy. Irish Acad. Conf., (1973), 89.

[12]

J. Jr. Douglas, T. Dupont and M. F. Wheeler, An $L$$infty$ estimate and a superconvergence result for a Galerkin method for elliptic equations based on tensor products of piecewise polynomials,, RAIRO Sér Rouge, 8 (1974), 61.

[13]

Y. Gong and X. Xiang, A class of optimal control problems of systems governed by the first order linear dynamic equations on time scales,, Journal of Industrial and Management Optimization (JIMO), 5 (2009), 1.

[14]

Paul B. Hermanns and Nguyen Van Thoai, Global optimization algorithm for solving bilevel programming problems with quadratic lower levels,, Journal of Industrial and Management Optimization (JIMO), 6 (2010), 177.

[15]

G. Knowles, Finite element approximation of parabolic time optimal control problems,, SIAM J. Control Optim., 20 (1982), 414. doi: 10.1137/0320032.

[16]

D. Kwak, S. Lee and Q. Li, Superconvergence of finite element method for parabolic problem,, Internal. J. Math. Sci., 23 (2000), 567. doi: 10.1155/S0161171200002519.

[17]

Y. Kwon and F. A. Milner, $L^\infty$-error estimates for mixed methods for semilinear second-order elliptic equations,, SIAM J. Numer. Anal., 25 (1988), 46. doi: 10.1137/0725005.

[18]

, R. Li and W. B. Liu,, Available from: \url{http://dsec.pku.edu.cn/~yuhj/computing/AFEPack/AFEPackIndex.html}., ().

[19]

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

[20]

J. L. Lions, "Optimal Control of Systems Governed by Partial Differential Equations,", Translated from the French by S. K. Mitter, (1971).

[21]

J. L. Lions and E. Magenes, "Non Homogeneous Boundary Value Problems and Applications,", Springer-Verlag, (1972).

[22]

Chongyang Liu, Zhaohua Gong and Enmin Feng, Modelling and optimal control for nonlinear multistage dynamical system of microbial fed-batch culture,, Journal of Industrial and Management Optimization (JIMO), 5 (2009), 835. doi: 10.3934/jimo.2009.5.835.

[23]

W. B. Liu and N. Yan, A Posteriori error estimates for optimal control problems governed by parabolic equations,, Numer. Math., 93 (2003), 497. doi: 10.1007/s002110100380.

[24]

Z. Lu and Y. Chen, A posteriori error estimates of triangular mixed finite element methods for semilinear optimal control problems,, Adv. Appl. Math. Mech., 1 (2009), 242.

[25]

Z. Lu and Y. Chen, $L$$infty$-error estimates of triangular mixed finite element methods for optimal control problem govern by semilinear elliptic equation,, Numer. Anal. Appl., 12 (2009), 74.

[26]

M. F. Wheeler, A priori $L^2$ error estimates for Galerkin approximation to parabolic partial differential equations,, SIAM J. Numer. Anal., 10 (1973), 723. doi: 10.1137/0710062.

[27]

R. S. Mcknight and W. E. Borsarge, The Ritz-Galerkin procedure for parabolic control problems,, SIAM J. Control, 11 (1973), 510. doi: 10.1137/0311040.

[28]

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

[29]

P. Neittaanmäki and D. Tiba, "Optimal Control of Nonlinear Parabolic Systems: Theory, Algorithms, and Applications,", Monographs and Textbooks in Pure and Applied Mathematics, 179 (1994).

[30]

Y. Y. Nie and V. Thomée, A lumped mass finite element method with quadrature for a nonlinear parabolic problem,, IMA J. Numer. Anal., 5 (1985), 371. doi: 10.1093/imanum/5.4.371.

[31]

L. A. Oganesjan and L. A. Ruhovec, An investigation of the rate of convergence of variation-difference schemes for second order elliptic equations in a two-dimensional region with smooth boundary,, Ž.Vyčisl. Mat. i Mat. Fiz, 9 (1969), 1102.

[32]

A. H. Schatz, I. H. Sloan and L. B. Wahlbin, Superconvergence in finite element methods and meshes that are locally symmetric with respect to a point,, SIAM J. Numer. Anal., 33 (1996), 505. doi: 10.1137/0733027.

[33]

V. Thomée, "Galerkin Finite Element Methods for Parabolic Problems," 2nd edition,, Springer Series in Compu. Math., 25 (2006).

[34]

V. Thomée, J. C. Xu and N. Y. Zhang, Superconvergence of the gradient in piecewise linear finite-element approximation to a parabolic problem,, SIAM J. Numer. Anal., 26 (1989), 553. doi: 10.1137/0726033.

[35]

F. Tröltzsch, Semidiscrete Ritz-Galerkin approximation of nonlinear parabolic boundary control problems-strong convergence of optimal control,, Appl. Math. Optim., 29 (1994), 309. doi: 10.1007/BF01189480.

[36]

L. Wahlbin, "Superconvergence in Gelerkin Finite Element Methods,", Lecture Notes in Math., 1605 (1995).

[37]

Changzhi Wu, Kok Lay Teo and Volker Rehbock, Optimal control of piecewise affine systems with piecewise affine state feedback,, Journal of Industrial and Management Optimization (JIMO), 5 (2009), 737.

[38]

X. Xing and Y. Chen, Error estimates of mixed methods for optimal control problems governed by parabolic equations,, Int. J. Numer. Methods Engineering, 75 (2008), 735. doi: 10.1002/nme.2289.

[39]

N. Yan, Superconvergence and recovery type a posteriori error estimate for constrained convex optimal control problems,, in, (2004), 408.

[40]

Changjun Yu, Kok Lay Teo, Liansheng Zhang and Yanqin Bai, A new exact penalty function method for continuous inequality constrained optimization problems,, Journal of Industrial and Management Optimization (JIMO), 6 (2010), 895. doi: 10.3934/jimo.2010.6.895.

[41]

C. D. Zhu and Q. Lin, "Youxianyuan Chaoshoulian Lilun," (Chinese) [The Hyperconvergence Theory of Finite Elements],, Hunan Science and Technology Publishing House, (1989).

show all references

References:
[1]

A. B. Andreev and R. D. Lazarov, Superconvergence of the gradient for quadratic triangular finite elements,, Numer. Methods PDEs, 4 (1988), 15.

[2]

J. H. Bramble and A. H. Schatz, Higher order local accuracy by averaging in the finite element method,, Math. Comp., 31 (1977), 94.

[3]

C. M. Chen and Y. Q. Huang, "High Accuracy Theory of Finite Element Methods,", Hunan Science and Technology Press, (1995).

[4]

C. M. Chen and V. Thomée, The lumped mass finite element method for a parabolic problem,, J. Austral. Math. Soc. Ser. B, 26 (1985), 329. doi: 10.1017/S0334270000004549.

[5]

Y. Chen, Superconvergence of quadratic optimal control problems by triangular mixed finite element methods,, Int. J. Numer. Methods Engineering, 75 (2008), 881. doi: 10.1002/nme.2272.

[6]

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.

[7]

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

[8]

Y. Chen and W. B. Liu, Error estimates and superconvergence of mixed finite element for quadratic optimal control,, Int. J. Numer. Anal. Model., 3 (2006), 311.

[9]

Y. Chen, N. Y. Yi and W. B. Liu, A Legendre-Galerkin spectral method for optimal control problems governed by elliptic equations,, SIAM J. Numer. Anal., 46 (2008), 2254. doi: 10.1137/070679703.

[10]

P. G. Ciarlet, "The Finite Element Method for Elliptic Problems,", Studies in Mathematics and its Applications, 4 (1978).

[11]

J. Jr. Douglas and T. Dupont, Some superconvergence results for Galerkin methods for the approximate solution of two-point boundary problems,, Topics in Numerical Analysis (Proc. Roy. Irish Acad. Conf., (1973), 89.

[12]

J. Jr. Douglas, T. Dupont and M. F. Wheeler, An $L$$infty$ estimate and a superconvergence result for a Galerkin method for elliptic equations based on tensor products of piecewise polynomials,, RAIRO Sér Rouge, 8 (1974), 61.

[13]

Y. Gong and X. Xiang, A class of optimal control problems of systems governed by the first order linear dynamic equations on time scales,, Journal of Industrial and Management Optimization (JIMO), 5 (2009), 1.

[14]

Paul B. Hermanns and Nguyen Van Thoai, Global optimization algorithm for solving bilevel programming problems with quadratic lower levels,, Journal of Industrial and Management Optimization (JIMO), 6 (2010), 177.

[15]

G. Knowles, Finite element approximation of parabolic time optimal control problems,, SIAM J. Control Optim., 20 (1982), 414. doi: 10.1137/0320032.

[16]

D. Kwak, S. Lee and Q. Li, Superconvergence of finite element method for parabolic problem,, Internal. J. Math. Sci., 23 (2000), 567. doi: 10.1155/S0161171200002519.

[17]

Y. Kwon and F. A. Milner, $L^\infty$-error estimates for mixed methods for semilinear second-order elliptic equations,, SIAM J. Numer. Anal., 25 (1988), 46. doi: 10.1137/0725005.

[18]

, R. Li and W. B. Liu,, Available from: \url{http://dsec.pku.edu.cn/~yuhj/computing/AFEPack/AFEPackIndex.html}., ().

[19]

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

[20]

J. L. Lions, "Optimal Control of Systems Governed by Partial Differential Equations,", Translated from the French by S. K. Mitter, (1971).

[21]

J. L. Lions and E. Magenes, "Non Homogeneous Boundary Value Problems and Applications,", Springer-Verlag, (1972).

[22]

Chongyang Liu, Zhaohua Gong and Enmin Feng, Modelling and optimal control for nonlinear multistage dynamical system of microbial fed-batch culture,, Journal of Industrial and Management Optimization (JIMO), 5 (2009), 835. doi: 10.3934/jimo.2009.5.835.

[23]

W. B. Liu and N. Yan, A Posteriori error estimates for optimal control problems governed by parabolic equations,, Numer. Math., 93 (2003), 497. doi: 10.1007/s002110100380.

[24]

Z. Lu and Y. Chen, A posteriori error estimates of triangular mixed finite element methods for semilinear optimal control problems,, Adv. Appl. Math. Mech., 1 (2009), 242.

[25]

Z. Lu and Y. Chen, $L$$infty$-error estimates of triangular mixed finite element methods for optimal control problem govern by semilinear elliptic equation,, Numer. Anal. Appl., 12 (2009), 74.

[26]

M. F. Wheeler, A priori $L^2$ error estimates for Galerkin approximation to parabolic partial differential equations,, SIAM J. Numer. Anal., 10 (1973), 723. doi: 10.1137/0710062.

[27]

R. S. Mcknight and W. E. Borsarge, The Ritz-Galerkin procedure for parabolic control problems,, SIAM J. Control, 11 (1973), 510. doi: 10.1137/0311040.

[28]

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

[29]

P. Neittaanmäki and D. Tiba, "Optimal Control of Nonlinear Parabolic Systems: Theory, Algorithms, and Applications,", Monographs and Textbooks in Pure and Applied Mathematics, 179 (1994).

[30]

Y. Y. Nie and V. Thomée, A lumped mass finite element method with quadrature for a nonlinear parabolic problem,, IMA J. Numer. Anal., 5 (1985), 371. doi: 10.1093/imanum/5.4.371.

[31]

L. A. Oganesjan and L. A. Ruhovec, An investigation of the rate of convergence of variation-difference schemes for second order elliptic equations in a two-dimensional region with smooth boundary,, Ž.Vyčisl. Mat. i Mat. Fiz, 9 (1969), 1102.

[32]

A. H. Schatz, I. H. Sloan and L. B. Wahlbin, Superconvergence in finite element methods and meshes that are locally symmetric with respect to a point,, SIAM J. Numer. Anal., 33 (1996), 505. doi: 10.1137/0733027.

[33]

V. Thomée, "Galerkin Finite Element Methods for Parabolic Problems," 2nd edition,, Springer Series in Compu. Math., 25 (2006).

[34]

V. Thomée, J. C. Xu and N. Y. Zhang, Superconvergence of the gradient in piecewise linear finite-element approximation to a parabolic problem,, SIAM J. Numer. Anal., 26 (1989), 553. doi: 10.1137/0726033.

[35]

F. Tröltzsch, Semidiscrete Ritz-Galerkin approximation of nonlinear parabolic boundary control problems-strong convergence of optimal control,, Appl. Math. Optim., 29 (1994), 309. doi: 10.1007/BF01189480.

[36]

L. Wahlbin, "Superconvergence in Gelerkin Finite Element Methods,", Lecture Notes in Math., 1605 (1995).

[37]

Changzhi Wu, Kok Lay Teo and Volker Rehbock, Optimal control of piecewise affine systems with piecewise affine state feedback,, Journal of Industrial and Management Optimization (JIMO), 5 (2009), 737.

[38]

X. Xing and Y. Chen, Error estimates of mixed methods for optimal control problems governed by parabolic equations,, Int. J. Numer. Methods Engineering, 75 (2008), 735. doi: 10.1002/nme.2289.

[39]

N. Yan, Superconvergence and recovery type a posteriori error estimate for constrained convex optimal control problems,, in, (2004), 408.

[40]

Changjun Yu, Kok Lay Teo, Liansheng Zhang and Yanqin Bai, A new exact penalty function method for continuous inequality constrained optimization problems,, Journal of Industrial and Management Optimization (JIMO), 6 (2010), 895. doi: 10.3934/jimo.2010.6.895.

[41]

C. D. Zhu and Q. Lin, "Youxianyuan Chaoshoulian Lilun," (Chinese) [The Hyperconvergence Theory of Finite Elements],, Hunan Science and Technology Publishing House, (1989).

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