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 and 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), 15-32.

[2]

J. H. Bramble and A. H. Schatz, Higher order local accuracy by averaging in the finite element method, Math. Comp., 31 (1977), 94-111.

[3]

C. M. Chen and Y. Q. Huang, "High Accuracy Theory of Finite Element Methods," Hunan Science and Technology Press, Changsha, 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), 329-354. 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), 881-898. doi: 10.1002/nme.2272.

[6]

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.

[7]

Y. Chen and Y. Dai, Superconvergence for optimal control problems governed by semi-linear elliptic equations, J. Sci. Comp., 39 (2009), 206-221. 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), 311-321.

[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), 2254-2275. doi: 10.1137/070679703.

[10]

P. G. Ciarlet, "The Finite Element Method for Elliptic Problems," Studies in Mathematics and its Applications, 4, North-Holland Publishing Co., Amsterdam-New York-Oxford, 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., Univ. Coll., Dublin, 1972), Academic Press, London, (1973), 89-92.

[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), 61-66.

[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), 1-10.

[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), 177-196.

[15]

G. Knowles, Finite element approximation of parabolic time optimal control problems, SIAM J. Control Optim., 20 (1982), 414-427. 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), 567-578. 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), 46-53. doi: 10.1137/0725005.

[18]

, R. Li and W. B. Liu, Available from: 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), 1321-1349. 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, Die Grundlehren der Mathematischen Wissenschaften, Band 170, Springer-Verlag, New York-Berlin, 1971.

[21]

J. L. Lions and E. Magenes, "Non Homogeneous Boundary Value Problems and Applications," Springer-Verlag, Berlin, 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), 835-850. 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), 497-521. 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), 242-256.

[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), 74-86.

[26]

M. F. Wheeler, A priori $L^2$ error estimates for Galerkin approximation to parabolic partial differential equations, SIAM J. Numer. Anal., 10 (1973), 723-759. 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), 510-524. doi: 10.1137/0311040.

[28]

C. Meyer and A. Rösch, Superconvergence properties of optimal control problems, SIAM J. Control Optim., 43 (2004), 970-985. 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, Marcel Dekker, Inc., New York, 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), 371-396. 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), 1102-1120.

[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), 505-521. doi: 10.1137/0733027.

[33]

V. Thomée, "Galerkin Finite Element Methods for Parabolic Problems," 2nd edition, Springer Series in Compu. Math., 25, Springer-Verlag, Berlin, 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), 553-573. 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), 309-329. doi: 10.1007/BF01189480.

[36]

L. Wahlbin, "Superconvergence in Gelerkin Finite Element Methods," Lecture Notes in Math., 1605, Springer-Verlag, Berlin, 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), 737-747.

[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), 735-754. doi: 10.1002/nme.2289.

[39]

N. Yan, Superconvergence and recovery type a posteriori error estimate for constrained convex optimal control problems, in "Adv. Sci. Comput. Appl." (eds. Y. Lu, W. Sun and T. Tang), Science Press, (2004), 408-419.

[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), 895-910. 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, Changsha, 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), 15-32.

[2]

J. H. Bramble and A. H. Schatz, Higher order local accuracy by averaging in the finite element method, Math. Comp., 31 (1977), 94-111.

[3]

C. M. Chen and Y. Q. Huang, "High Accuracy Theory of Finite Element Methods," Hunan Science and Technology Press, Changsha, 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), 329-354. 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), 881-898. doi: 10.1002/nme.2272.

[6]

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.

[7]

Y. Chen and Y. Dai, Superconvergence for optimal control problems governed by semi-linear elliptic equations, J. Sci. Comp., 39 (2009), 206-221. 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), 311-321.

[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), 2254-2275. doi: 10.1137/070679703.

[10]

P. G. Ciarlet, "The Finite Element Method for Elliptic Problems," Studies in Mathematics and its Applications, 4, North-Holland Publishing Co., Amsterdam-New York-Oxford, 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., Univ. Coll., Dublin, 1972), Academic Press, London, (1973), 89-92.

[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), 61-66.

[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), 1-10.

[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), 177-196.

[15]

G. Knowles, Finite element approximation of parabolic time optimal control problems, SIAM J. Control Optim., 20 (1982), 414-427. 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), 567-578. 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), 46-53. doi: 10.1137/0725005.

[18]

, R. Li and W. B. Liu, Available from: 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), 1321-1349. 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, Die Grundlehren der Mathematischen Wissenschaften, Band 170, Springer-Verlag, New York-Berlin, 1971.

[21]

J. L. Lions and E. Magenes, "Non Homogeneous Boundary Value Problems and Applications," Springer-Verlag, Berlin, 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), 835-850. 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), 497-521. 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), 242-256.

[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), 74-86.

[26]

M. F. Wheeler, A priori $L^2$ error estimates for Galerkin approximation to parabolic partial differential equations, SIAM J. Numer. Anal., 10 (1973), 723-759. 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), 510-524. doi: 10.1137/0311040.

[28]

C. Meyer and A. Rösch, Superconvergence properties of optimal control problems, SIAM J. Control Optim., 43 (2004), 970-985. 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, Marcel Dekker, Inc., New York, 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), 371-396. 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), 1102-1120.

[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), 505-521. doi: 10.1137/0733027.

[33]

V. Thomée, "Galerkin Finite Element Methods for Parabolic Problems," 2nd edition, Springer Series in Compu. Math., 25, Springer-Verlag, Berlin, 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), 553-573. 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), 309-329. doi: 10.1007/BF01189480.

[36]

L. Wahlbin, "Superconvergence in Gelerkin Finite Element Methods," Lecture Notes in Math., 1605, Springer-Verlag, Berlin, 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), 737-747.

[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), 735-754. doi: 10.1002/nme.2289.

[39]

N. Yan, Superconvergence and recovery type a posteriori error estimate for constrained convex optimal control problems, in "Adv. Sci. Comput. Appl." (eds. Y. Lu, W. Sun and T. Tang), Science Press, (2004), 408-419.

[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), 895-910. 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, Changsha, 1989.

[1]

Eduardo Casas, Mariano Mateos, Arnd Rösch. Finite element approximation of sparse parabolic control problems. Mathematical Control and Related Fields, 2017, 7 (3) : 393-417. doi: 10.3934/mcrf.2017014

[2]

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

[3]

Philip Trautmann, Boris Vexler, Alexander Zlotnik. Finite element error analysis for measure-valued optimal control problems governed by a 1D wave equation with variable coefficients. Mathematical Control and Related Fields, 2018, 8 (2) : 411-449. doi: 10.3934/mcrf.2018017

[4]

Dongho Kim, Eun-Jae Park. Adaptive Crank-Nicolson methods with dynamic finite-element spaces for parabolic problems. Discrete and Continuous Dynamical Systems - B, 2008, 10 (4) : 873-886. doi: 10.3934/dcdsb.2008.10.873

[5]

Lijuan Wang, Jun Zou. Error estimates of finite element methods for parameter identifications in elliptic and parabolic systems. Discrete and Continuous Dynamical Systems - B, 2010, 14 (4) : 1641-1670. doi: 10.3934/dcdsb.2010.14.1641

[6]

Xiaomeng Li, Qiang Xu, Ailing Zhu. Weak Galerkin mixed finite element methods for parabolic equations with memory. Discrete and Continuous Dynamical Systems - S, 2019, 12 (3) : 513-531. doi: 10.3934/dcdss.2019034

[7]

Ming Yan, Lili Chang, Ningning Yan. Finite element method for constrained optimal control problems governed by nonlinear elliptic PDEs. Mathematical Control and Related Fields, 2012, 2 (2) : 183-194. doi: 10.3934/mcrf.2012.2.183

[8]

Zhangxin Chen. On the control volume finite element methods and their applications to multiphase flow. Networks and Heterogeneous Media, 2006, 1 (4) : 689-706. doi: 10.3934/nhm.2006.1.689

[9]

Qun Lin, Hehu Xie. Recent results on lower bounds of eigenvalue problems by nonconforming finite element methods. Inverse Problems and Imaging, 2013, 7 (3) : 795-811. doi: 10.3934/ipi.2013.7.795

[10]

Bin Wang, Lin Mu. Viscosity robust weak Galerkin finite element methods for Stokes problems. Electronic Research Archive, 2021, 29 (1) : 1881-1895. doi: 10.3934/era.2020096

[11]

Ugur G. Abdulla. On the optimal control of the free boundary problems for the second order parabolic equations. II. Convergence of the method of finite differences. Inverse Problems and Imaging, 2016, 10 (4) : 869-898. doi: 10.3934/ipi.2016025

[12]

Karl Kunisch, Lijuan Wang. Bang-bang property of time optimal controls of semilinear parabolic equation. Discrete and Continuous Dynamical Systems, 2016, 36 (1) : 279-302. doi: 10.3934/dcds.2016.36.279

[13]

Tao Lin, Yanping Lin, Weiwei Sun. Error estimation of a class of quadratic immersed finite element methods for elliptic interface problems. Discrete and Continuous Dynamical Systems - B, 2007, 7 (4) : 807-823. doi: 10.3934/dcdsb.2007.7.807

[14]

Zuliang Lu, Fei Huang, Xiankui Wu, Lin Li, Shang Liu. Convergence and quasi-optimality of $ L^2- $norms based an adaptive finite element method for nonlinear optimal control problems. Electronic Research Archive, 2020, 28 (4) : 1459-1486. doi: 10.3934/era.2020077

[15]

Jinghong Liu, Yinsuo Jia. Gradient superconvergence post-processing of the tensor-product quadratic pentahedral finite element. Discrete and Continuous Dynamical Systems - B, 2015, 20 (2) : 495-504. doi: 10.3934/dcdsb.2015.20.495

[16]

Dominik Hafemeyer, Florian Mannel, Ira Neitzel, Boris Vexler. Finite element error estimates for one-dimensional elliptic optimal control by BV-functions. Mathematical Control and Related Fields, 2020, 10 (2) : 333-363. doi: 10.3934/mcrf.2019041

[17]

Z. Foroozandeh, Maria do rosário de Pinho, M. Shamsi. On numerical methods for singular optimal control problems: An application to an AUV problem. Discrete and Continuous Dynamical Systems - B, 2019, 24 (5) : 2219-2235. doi: 10.3934/dcdsb.2019092

[18]

Martin Benning, Elena Celledoni, Matthias J. Ehrhardt, Brynjulf Owren, Carola-Bibiane Schönlieb. Deep learning as optimal control problems: Models and numerical methods. Journal of Computational Dynamics, 2019, 6 (2) : 171-198. doi: 10.3934/jcd.2019009

[19]

Xiaowei Pang, Haiming Song, Xiaoshen Wang, Jiachuan Zhang. Efficient numerical methods for elliptic optimal control problems with random coefficient. Electronic Research Archive, 2020, 28 (2) : 1001-1022. doi: 10.3934/era.2020053

[20]

Alexander Arguchintsev, Vasilisa Poplevko. An optimal control problem by parabolic equation with boundary smooth control and an integral constraint. Numerical Algebra, Control and Optimization, 2018, 8 (2) : 193-202. doi: 10.3934/naco.2018011

2021 Impact Factor: 1.411

Metrics

  • PDF downloads (76)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]