American Institute of Mathematical Sciences

Advanced
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

[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

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

[1]

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
[2]

Dominik Hafemeyer, Florian Mannel, Ira Neitzel, Boris Vexler. Finite element error estimates for one-dimensional elliptic optimal control by BV-functions. Mathematical Control & Related Fields, 2019, 0 (0) : 0-0. doi: 10.3934/mcrf.2019041
[3]

Xiaomeng Li, Qiang Xu, Ailing Zhu. Weak Galerkin mixed finite element methods for parabolic equations with memory. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 513-531. doi: 10.3934/dcdss.2019034
[4]

Ming Yan, Lili Chang, Ningning Yan. Finite element method for constrained optimal control problems governed by nonlinear elliptic PDEs. Mathematical Control & Related Fields, 2012, 2 (2) : 183-194. doi: 10.3934/mcrf.2012.2.183
[5]

Martin Burger, José A. Carrillo, Marie-Therese Wolfram. A mixed finite element method for nonlinear diffusion equations. Kinetic & Related Models, 2010, 3 (1) : 59-83. doi: 10.3934/krm.2010.3.59
[6]

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

Eduardo Casas, Mariano Mateos, Arnd Rösch. Finite element approximation of sparse parabolic control problems. Mathematical Control & Related Fields, 2017, 7 (3) : 393-417. doi: 10.3934/mcrf.2017014
[8]

Wolf-Jüergen Beyn, Janosch Rieger. Galerkin finite element methods for semilinear elliptic differential inclusions. Discrete & Continuous Dynamical Systems - B, 2013, 18 (2) : 295-312. doi: 10.3934/dcdsb.2013.18.295
[9]

Lijuan Wang, Jun Zou. Error estimates of finite element methods for parameter identifications in elliptic and parabolic systems. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1641-1670. doi: 10.3934/dcdsb.2010.14.1641
[10]

Zhangxin Chen. On the control volume finite element methods and their applications to multiphase flow. Networks & Heterogeneous Media, 2006, 1 (4) : 689-706. doi: 10.3934/nhm.2006.1.689
[11]

Qun Lin, Hehu Xie. Recent results on lower bounds of eigenvalue problems by nonconforming finite element methods. Inverse Problems & Imaging, 2013, 7 (3) : 795-811. doi: 10.3934/ipi.2013.7.795
[12]

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 & Related Fields, 2018, 8 (2) : 411-449. doi: 10.3934/mcrf.2018017
[13]

Dongho Kim, Eun-Jae Park. Adaptive Crank-Nicolson methods with dynamic finite-element spaces for parabolic problems. Discrete & Continuous Dynamical Systems - B, 2008, 10 (4) : 873-886. doi: 10.3934/dcdsb.2008.10.873
[14]

Jinghong Liu, Yinsuo Jia. Gradient superconvergence post-processing of the tensor-product quadratic pentahedral finite element. Discrete & Continuous Dynamical Systems - B, 2015, 20 (2) : 495-504. doi: 10.3934/dcdsb.2015.20.495
[15]

G. Bonanno, Salvatore A. Marano. Highly discontinuous elliptic problems. Conference Publications, 1998, 1998 (Special) : 118-123. doi: 10.3934/proc.1998.1998.118
[16]

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 & Imaging, 2016, 10 (4) : 869-898. doi: 10.3934/ipi.2016025
[17]

Assyr Abdulle, Yun Bai, Gilles Vilmart. Reduced basis finite element heterogeneous multiscale method for quasilinear elliptic homogenization problems. Discrete & Continuous Dynamical Systems - S, 2015, 8 (1) : 91-118. doi: 10.3934/dcdss.2015.8.91
[18]

Mahboub Baccouch. Superconvergence of the semi-discrete local discontinuous Galerkin method for nonlinear KdV-type problems. Discrete & Continuous Dynamical Systems - B, 2019, 24 (1) : 19-54. doi: 10.3934/dcdsb.2018104
[19]

Z. Foroozandeh, Maria do rosário de Pinho, M. Shamsi. On numerical methods for singular optimal control problems: An application to an AUV problem. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2219-2235. doi: 10.3934/dcdsb.2019092
[20]

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

2018 Impact Factor: 1.025

Tools

Metrics

  • PDF downloads (20)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences