Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-24T06:03:20.667Z Has data issue: false hasContentIssue false
  • English
  • Français

A Priori Error Estimate of Splitting Positive Definite Mixed Finite Element Method for Parabolic Optimal Control Problems

Part of: Optimality conditions Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  24 May 2016

Hongfei Fu ,
Hongxing Rui ,
Jiansong Zhang  and
Hui Guo
Hongfei Fu*
Affiliation:
College of Science, China University of Petroleum, Qingdao, 266580, China
Hongxing Rui*
Affiliation:
School of Mathematics, Shandong University, Jinan 250100, China
Jiansong Zhang*
Affiliation:
College of Science, China University of Petroleum, Qingdao, 266580, China
Hui Guo*
Affiliation:
College of Science, China University of Petroleum, Qingdao, 266580, China
*
*Corresponding author. Email addresses:[email protected](H. Fu), [email protected](H. Rui), [email protected](J. Zhang), [email protected](H. Guo)
*Corresponding author. Email addresses:[email protected](H. Fu), [email protected](H. Rui), [email protected](J. Zhang), [email protected](H. Guo)
*Corresponding author. Email addresses:[email protected](H. Fu), [email protected](H. Rui), [email protected](J. Zhang), [email protected](H. Guo)
*Corresponding author. Email addresses:[email protected](H. Fu), [email protected](H. Rui), [email protected](J. Zhang), [email protected](H. Guo)
Get access

Abstract

In this paper, we propose a splitting positive definite mixed finite element method for the approximation of convex optimal control problems governed by linear parabolic equations, where the primal state variable y and its flux σ are approximated simultaneously. By using the first order necessary and sufficient optimality conditions for the optimization problem, we derive another pair of adjoint state variables z and ω, and also a variational inequality for the control variable u is derived. As we can see the two resulting systems for the unknown state variable y and its flux σ are splitting, and both symmetric and positive definite. Besides, the corresponding adjoint states z and ω are also decoupled, and they both lead to symmetric and positive definite linear systems. We give some a priori error estimates for the discretization of the states, adjoint states and control, where Ladyzhenkaya-Babuska-Brezzi consistency condition is not necessary for the approximation of the state variable y and its flux σ. Finally, numerical experiments are given to show the efficiency and reliability of the splitting positive definite mixed finite element method.

Keywords

Parabolic optimal controlsplitting mixed finite element methodpositive definitea priori error estimatesnumerical experiments

MSC classification

Secondary: 49K20: Problems involving partial differential equations 65M60: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 9 , Issue 2 , May 2016 , pp. 215 - 238
Copyright
Copyright © Global-Science Press 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Geveci, T., On the approximation of the solution of an optimal control problem governed by an elliptic equation, RAIRO Anal. Numer., 13 (1979), pp. 313–328.Google Scholar
[2]Liu, W. and Tiba, D., Error estimates in the approximation of optimization problems governed by nonlinear operators, Numer. Func. Anal. Optim., 22 (2001), pp. 953–972.Google Scholar
[3]Rösch, A., Error estimates for linear-quadratic control problems with control constraints, Optim. Methods Softw., 21 (2005), pp. 121–134.Google Scholar
[4]Zhou, J., Chen, Y., and Dai, Y., Superconvergence of triangular mixed finite elements for optimal control problems with an integral constraint, Appl. Math. Comput., 217 (2010), pp. 2057–2066.Google Scholar
[5]Chen, Y., Huang, F., Yi, N., and Liu, W., A Legendre-Galerkin spectral method for optimal control problems governed by Stokes equations, SIAM J. Numer. Anal., 49 (2011), pp. 1625–1648.Google Scholar
[6]Xing, X. and Chen, Y., Error estimates of mixed methods for optimal control problems governed by parabolic equations, Int. J. Numer. Methods Engrg., 75 (2008), pp. 735–754.CrossRefGoogle Scholar
[7]Meidner, D. and Vexler, B., A priori error estimates for space-time finite element discretization of parabolic optimal control problems. Part I: problems without control constraints, SIAM J. Control Optim., 47 (2008), pp. 1150–1177.Google Scholar
[8]Meidner, D. and Vexler, B., A priori error estimates for space-time finite element discretization of parabolic optimal control problems. Part II: problems with control constraints, SIAM J. Control Optim., 47 (2008), pp. 1301–1329.Google Scholar
[9]Chen, Y. and Lu, Z., Error estimates of fully discrete mixed finite element methods for semilinear quadratic parabolic optimal control problem, Comput. Methods Appl. Mech. Engrg., 199 (2010), pp. 1415–1423.Google Scholar
[10]Fu, H. and Rui, H., A priori error estimates for optimal control problems governed by transient advection-diffusion equations, J. Sci. Comput., 38 (2009), pp. 290–315.Google Scholar
[11]Fu, H. and Rui, H., A characteristic-mixed finite element method for time-dependent convection-diffusion optimal control problem, Appl.Math. Comput., 218 (2011), pp. 3430–3440.Google Scholar
[12]Becker, R., Kapp, H., and Rannacher, R., Adaptive finite element methods for optimal control of partial differential equations: Basic concept, SIAM J. Control Optim., 39 (2000), pp. 113–132.Google Scholar
[13]Liu, W. and Yan, N., A posteriori error estimates for distributed convex optimal control problems, Advance. Comput. Math., 15 (2001), pp. 285–309.Google Scholar
[14]Li, R., Liu, W., Ma, H., and Tang, T., Adaptive finite elememt approximation of elliptic optimal control, SIAM J. Control Optim., 41 (2002), pp.1321–1349.Google Scholar
[15]Liu, W. and Yan, N., A posteriori error estimates for control problems governed by Stokes equations, SIAM J. Numer. Anal., 40 (2002), pp. 1850–1869.Google Scholar
[16]Liu, W. and Yan, N., A posteriori error estimates for optimal control problems governed by parabolic equations, Numer. Math., 93 (2003), pp. 497–521.Google Scholar
[17]Gong, W. and Yan, N., A posteriori error estimate for boundary control problems governed by the parabolic partial differential equations, J. Comput. Math., 27 (2009), pp. 68–88.Google Scholar
[18]Brenner, S. C. and Scott, L. R., The Mathematical Theory of Finite Element Methods (3rd ed.), Springer-Verlag, Berlin, 2008.CrossRefGoogle Scholar
[19]Ciarlet, P. G., The Finite Element Method for Elliptic Problems, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2002.Google Scholar
[20]Evans, L. C., Partial Differential Equations, vol. 19 of Grad. Stud. Math., AMS, Providence, 2002.Google Scholar
[21]Lions, J. L., Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, Berlin, 1971.Google Scholar
[22]Pironneau, O., Optimal Shape Design for Elliptic Systems, Springer-Verlag, Berlin, 1984.Google Scholar
[23]Tiba, D., Lectures on the Optimal Control of Elliptic Equations, University of Jyvaskyla Press, Finland, 1995.Google Scholar
[24]Liu, W. and Yan, N., Adaptive Finite Element Methods for Optimal Control Governed by PDEs, Series in Information and Computational Science 41, Science Press, Beijing, 2008.Google Scholar
[25]Kufner, A., John, O., and Fucik, S., Function Spaces, Nordhoff, Leyden, 1977.Google Scholar
[26]Li, R. and Liu, W., http://circus.math.pku.edu.cn/AFEPack.Google Scholar
[27]Raviart Anf, P. A.Thomas, J. M., A mixed finite element method for 2nd order elliptic problems, Lecture Notes in Mathematics, vol. 606, Springer, Berlin, 1977, pp. 292–315.Google Scholar
[28]Nedelec, J. C., Mixed finite element in R3, Numer. Math., 35 (1980), pp. 315–341.Google Scholar
[29]Brezzi, F., Douglas, J. Jr., Duran, R., and Marini, L. D., Mixed finite elements for second order elliptic problems in three space variables, Numer. Math., 51 (1987), pp. 237–250.Google Scholar
[30]Brezzi, F., Douglas, J. Jr., Fortin, M., and Marini, L. D., Efficient rectangular mixed finite elements in two and three space variables, Rairo Model Math. Numer. Anal., 21 (1987), pp. 581–603.Google Scholar
[31]Brezzi, F., Douglas, J. Jr., and Marini, L. D., Two family of mixed finite elements for second order elliptic problems, Numer. Math., 47 (1985), pp. 217–235.Google Scholar
[32]Thomée, V., Galerkin Finite Element Methods for Parabolic Problems (2nd ed.), Springer Series in Computational Mathematics, Springer-Verlag, Berlin, 2006.Google Scholar
[33]Wheeler, M.F., A priori L2 error estimates for Galerkin approximations to parabolic partial differential equations, SIAM J. Numer. Anal., 10 (1973), pp. 723–759.Google Scholar