# American Institute of Mathematical Sciences

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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