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.   Google Scholar

[2]

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

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[10]

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

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[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.  Google Scholar

[20]

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

[21]

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

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

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

[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).   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[33]

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

[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.  Google Scholar

[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.  Google Scholar

[36]

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

[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.   Google Scholar

[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.  Google Scholar

[39]

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

[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.  Google Scholar

[41]

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

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.   Google Scholar

[2]

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

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[10]

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

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[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.  Google Scholar

[20]

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

[21]

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

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

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

[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).   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[33]

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

[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.  Google Scholar

[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.  Google Scholar

[36]

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

[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.   Google Scholar

[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.  Google Scholar

[39]

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

[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.  Google Scholar

[41]

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

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