JOURNAL BROWSE
Search
Advanced SearchSearch Tips
RICHARDSON EXTRAPOLATION AND DEFECT CORRECTION OF MIXED FINITE ELEMENT METHODS FOR ELLIPTIC OPTIMAL CONTROL PROBLEMS
facebook(new window)  Pirnt(new window) E-mail(new window) Excel Download
 Title & Authors
RICHARDSON EXTRAPOLATION AND DEFECT CORRECTION OF MIXED FINITE ELEMENT METHODS FOR ELLIPTIC OPTIMAL CONTROL PROBLEMS
Chen, Yanping; Huang, Yunqing; Hou, Tianliang;
  PDF(new window)
 Abstract
In this paper asymptotic error expansions for mixed finite element approximations to a class of second order elliptic optimal control problems are derived under rectangular meshes, and the Richardson extrapolation of two different schemes and interpolation defect correction can be applied to increase the accuracy of the approximations. As a by-product, we illustrate that all the approximations of higher accuracy can be used to form a class of a posteriori error estimators of the mixed finite element method for optimal control problems.
 Keywords
optimal control problems;mixed finite element methods;asymptotic expansions;interpolation postprocessing;defect correction;a posteriori error estimators;
 Language
English
 Cited by
 References
1.
W. Alt, On the approximation of infinite optimization problems with an application to optimal control problems, Appl. Math. Optim. 12 (1984), no. 1, 15-27. crossref(new window)

2.
W. Alt and U. Machenroth, Convergence of finite element approximations to state constrained convex parabolic boundary control problems, SIAM J. Control Optim. 27 (1989), no. 4, 718-736. crossref(new window)

3.
N. Arada, E. Casas, and F. Troltzsch, Error estimates for the numerical approximation of a semilinear elliptic control problem, Comput. Optim. Appl. 23 (2002), no. 2, 201- 229. crossref(new window)

4.
H. Blum, Q. Lin, and R. Rannacher, Asymptotic error expansion and Richardson ex- trapolation for linear finite elements, Numer. Math. 49 (1986), no. 1, 11-37. crossref(new window)

5.
H. Brunner, Y. Lin, and S. Zhang, Higher accuracy methods for second-kind Volterra in- tegral equations based on asymptotic expansions of iterated Galerkin methods, J. Integral Equations Appl. 10 (1998), no. 4, 375-396. crossref(new window)

6.
C. Chen and Y. Huang, Higher Accuracy Theory of FEM, Hunan Science Press, Changsha, China, 1995.

7.
Y. Chen, Superconvergence of mixed finite element methods for optimal control problems, Math. Comp. 77 (2008), no. 263, 1269-1291. crossref(new window)

8.
Y. Chen, Superconvergence of quadratic optimal control problems by triangular mixed finite elements, Internat. J. Numer. Methods Engrg. 75 (2008), no. 8, 881-898. crossref(new window)

9.
Y. Chen and W. B. Liu, Error estimates and superconvergence of mixed finite elements for quadratic optimal control, Int. J. Numer. Anal. Model. 3 (2006), no. 3, 311-321.

10.
Y. Chen and W. B. Liu, A posteriori error estimates for mixed finite element solutions of convex optimal control problems, J. Comput. Appl. Math. 211 (2008), no. 1, 76-89. crossref(new window)

11.
Y. Chen and Z. Lu, Error estimates of fully discrete mixed finite element methods for semilinear quadratic parabolic optimal control problem, Comput. Methods Appl. Mech. Engrg. 199 (2010), no. 23-24, 1415-1423. crossref(new window)

12.
I. Chryssoverghi, Discretization methods for semilinear parabolic optimal control problems, Int. J. Numer. Anal. Model. 3 (2006), no. 4, 437-458.

13.
J. Douglas and J. E. Roberts, Global estimates for mixed methods for second order elliptic equations, Math. Comp. 44 (1985), no. 169, 39-52. crossref(new window)

14.
R. E. Ewing, Y. Lin, T. Sun, J. Wang, and S. Zhang, Sharp $L^{2}$-error estimates and superconvergence of mixed finite element methods for non-Fickian ows in porous media, SIAM J. Numer. Anal. 40 (2002), no. 4, 1538-1560. crossref(new window)

15.
G. Fairweather, Q. Lin, Y. Lin, J. Wang, and S. Zhang, Asymptotic expansions and Richardson extrapolation of approximate solutions for second order elliptic problems on rectangular domains by mixed finite element methods, SIAM J. Numer. Anal. 44 (2006), no. 3, 1122-1149. crossref(new window)

16.
M. D. Gunzburger and S. L. Hou, Finite-dimensional approximation of a class of constrained nonlinear optimal control problems, SIAM J. Control Optim. 34 (1996), no. 3, 1001-1043. crossref(new window)

17.
J. Haslinger and P. Neittaanmaki, Finite Element Approximation for Optimal Shape Design, John Wiley and Sons, Chichester, UK, 1988.

18.
P. Helfrich, Asymptotic expansion for the finite element approximation of parabolic problems, Extrapolation procedures in the finite element method (Bonn, 1983), 11-30, Bonner Math. Schriften, 158, Univ. Bonn, Bonn, 1984.

19.
L. Hou and J. C. Turner, Analysis and finite element approximation of an optimal control problem in electrochemistry with current density controls, Numer. Math. 71 (1995), no. 3, 289-315. crossref(new window)

20.
G. Knowles, Finite element approximation of parabolic time optimal control problems, SIAM J. Control Optim. 20 (1982), no. 3, 414-427. crossref(new window)

21.
Q. Lin, I. H. Sloan, and R. Xie, Extrapolation of the iterated-collocation method for integral equations, SIAM J. Numer. Anal. 27 (1990), no. 6, 1535-1541. crossref(new window)

22.
Q. Lin and N. Yan, The Construction and Analysis of High Efficiency Finite Element Methods, Hebei University Press, Baoding, China, 1996.

23.
Q. Lin, S. Zhang, and N. Yan, Extrapolation and defect correction for diffusion equations with boundary integral conditions, Acta Math. Sci. Ser. B Engl. Ed. 17 (1997), no. 4, 405-412.

24.
Q. Lin, S. Zhang, and N. Yan, High accuracy analysis for integrodifferential equations, Acta Math. Appl. Sin. Engl. Ser. 14 (1998), no. 2, 202-211. crossref(new window)

25.
Q. Lin, S. Zhang, and N. Yan, Methods for improving approximate accuracy for hyperbolic integrodifferential equations, J. Systems Sci. Math. Sci. 10 (1997), no. 3, 282-288.

26.
T. Lin, Y. Lin, M. Rao, and S. Zhang, Petrov-Galerkin methods for linear Volterra integro-differential equations, SIAM J. Numer. Anal. 38 (2000), no. 3, 937-963. crossref(new window)

27.
J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer, Berlin, 1971.

28.
H. Liu and N. Yan, Global Superconvergence for optimal control problems governed by Stokes equations, Int. J. Numer. Anal. Model. 3 (2006), no. 3, 283-302.

29.
T. Liu, N. Yan, and S. Zhang, Richardson extrapolation and defect correction of finite element methods for optimal control problems, J. Comput. Math. 28 (2010), no. 1, 55-71.

30.
R. S. Mcknight and W. E. Borsarge, The Ritz-Galerkin procedure for parabolic control problems, SIAM J. Control Optim. 11 (1973), 510-524. crossref(new window)

31.
P. Neittaanmaki and D. Tiba, Optimal Control of Nonlinear Parabolic Systems: Theory, Algorithms and Applications, M. Dekker, New York, 1994.

32.
D. Tiba, Lectures on the Optimal Control of Elliptic Problems, University of Jyvaskyla Press, Jyvaskyla, Finland, 1995.

33.
F. Troltzsch, Semidiscrete Ritz-Galerkin approximation of nonlinear parabolic boundary control problems-strong convergence of optimal control, Appl. Math. Optim. 29 (1994), no. 3, 309-329. crossref(new window)

34.
J. Wang, Superconvergence and extrapolation for mixed finite element methods on rectangular domains, Math. Comp. 56 (1991), no. 194, 477-503. crossref(new window)

35.
J. Wang, Asymptotic expansions and $L^{\infty}$-error estimates for mixed finite element meth- ods for second order elliptic problems, Numer. Math. 55 (1989), no. 4, 401-430. crossref(new window)

36.
X. Xing and Y. Chen, Error estimates of mixed methods for optimal control problems governed by parabolic equations, Internat. J. Numer. Methods Engrg. 75 (2008), no. 6, 735-754. crossref(new window)

37.
N. Yan and K. Li, An extrapolation method for BEM, J. Comput. Math. 2 (1989), no. 2, 217-224.

38.
S. Zhang, T. Lin, Y. Lin, and M. Rao, Extrapolation and a-posteriori error estimators of Petrov-Galerkin methods for non-linear Volterra integro-differential equations, J. Comput. Math. 19 (2001), no. 4, 407-422.