A POSTERIORI L(L2)-ERROR ESTIMATES OF SEMIDISCRETE MIXED FINITE ELEMENT METHODS FOR HYPERBOLIC OPTIMAL CONTROL PROBLEMS

Title & Authors
A POSTERIORI L(L2)-ERROR ESTIMATES OF SEMIDISCRETE MIXED FINITE ELEMENT METHODS FOR HYPERBOLIC OPTIMAL CONTROL PROBLEMS
Hou, Tianliang;

Abstract
In this paper, we discuss the a posteriori error estimates of the semidiscrete mixed finite element methods for quadratic optimal control problems governed by linear hyperbolic equations. The state and the co-state are discretized by the order $\small{k}$ Raviart-Thomas mixed finite element spaces and the control is approximated by piecewise polynomials of order $\small{k(k{\geq}0)}$. Using mixed elliptic reconstruction method, a posterior $\small{L^{\infty}(L^2)}$-error estimates for both the state and the control approximation are derived. Such estimates, which are apparently not available in the literature, are an important step towards developing reliable adaptive mixed finite element approximation schemes for the control problem.
Keywords
a posteriori error estimates;optimal control problems;hyperbolic equations;elliptic reconstruction;semidiscrete mixed finite element methods;
Language
English
Cited by
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3.
A posteriori error estimates of mixed finite element solutions for fourth order parabolic control problems, Journal of Inequalities and Applications, 2015, 2015, 1
References
1.
S. Adjerid, A posteriori finite element error estimation for second-order hyperbolic problems, Comput. Methods Appl. Mech. Engry. 191 (2002), no. 41-42, 4699-4719.

2.
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.

3.
I. Babuska, M. Feistauer, and P. Solin, On one approach to a posteriori error estimates for evolution problems solved by the method of lines, Numer. Math. 89 (2001), no. 2, 225-256.

4.
I. Babuska and S. Ohnimus, A posteriori error estimation for the semidiscrete finite element method of parabolic partial differential equations, Comput. Methods Appl. Mech. Engry. 190 (2001), no. 35-36, 4691-4712.

5.
I. Babuska and T. Strouboulis, The Finite Element Method and its Reliability, Oxford University press, Oxford, 2001.

6.
G. A. Baker, Error estimates for finite element methods for second order hyperbolic equations, SIAM J. Numer. Anal. 13 (1976), no. 4, 564-576.

7.
G. A. Baker and J. H. Bramble, Semidiscrete and single step fully discrete approximations for second order hyperbolic equations, RAIRO Anal Numer. 13 (1979), no. 2, 75-100.

8.
G. A. Baker and V. A. Dougalis, On the $L^{\infty}-convergence$ of Galerkin approximations for second order hyperbolic equations, Math. Comp. 34 (1980), no. 150, 401-424.

9.
W. Bangerth and R. Rannacher, Finite element approximation of the acoustic wave equation: error control and mesh adaptation, East-West J. Numer. Math. 7 (1999), no. 4, 263-282.

10.
W. Bangerth and R. Rannacher, Adaptive finite element techniques for the acoustic wave equation, J. Comput. Acoust. 9 (2001), no. 2, 575-591.

11.
E. Becache, P. Joly, and C. Tsogka, An analysis of new mixed finite elements for the approximation of wave propagation problems, SIAM J. Numer. Anal. 37 (2000), no. 4, 1053-1084.

12.
C. Bernardi and E. Suli, Time and Space adaptivity for the second-order wave equation, Math. Models Methods Anal. Sci. 15 (2005), no. 2, 199-225.

13.
F. Brezzi and M. Fortin, Mixed and hybrid finite element methods, Springer Series in Computational Mathematics, 15. Springer-Verlag, New York, 1991.

14.
H. Brunner and N. Yan, Finite element methods for optimal control problems governed by integral equations and integro-differential equations, Numer. Math. 101 (2005), no. 1, 1-27.

15.
C. Carstensen, A posteriori error estimate for the mixed finite element method, Math. Comp. 66 (1997), no. 218, 465-476.

16.
Y. Chen, Superconvergence of quadratic optimal control problems by triangular mixed finite elements, Internat. J. Numer. Methods Engrg. 75 (2008), no. 8, 881-898.

17.
Y. Chen, Y. Huang, W. B. Liu, and N. N. Yan, Error estimates and superconvergence of mixed finite element methods for convex optimal control problems, J. Sci. Comput. 42 (2010), no. 3, 382-403.

18.
Y. Chen and W. B. Liu, A posteriori error estimates for mixed finite element solutions of convex optimal control problems, J. Comp. Appl. Math. 211 (2008), no. 1, 76-89.

19.
A. Demlow, O. Lakkis, and C. Makridakis, A posteriori error estimates in the maximum norm for parabolic problems, SIAM J. Numer. Anal. 47 (2009), no. 3, 2157-2176.

20.
V. A. Dougalis and S. M. Serbin, On the efficiency of some fully discrete Galerkin methods for second-order hyperbolic equations, Comput. Math. Appl. 7 (1981), no. 3, 261-279.

21.
J. Douglas and J. E. Roberts, Global estimates for mixed methods for second order elliptic equations, Math. Comp. 44 (1985), no. 169, 39-52.

22.
K. Eriksson and C. Johnson, Adaptive finite elements methods for parabolic problems I: A linear model problem, SIAM J. Numer. Anal. 28 (1991), no. 1, 43-77.

23.
W. Gong and N. Yan, A posteriori error estimate for boundary control problems gov- erned by the parabolic partial differential equations, J. Comput. Math. 27 (2009), no. 1, 68-88.

24.
J. Haslinger and P. Neittaanmaki, Finite Element Approximation for Optimal Shape Design, John Wiley and Sons, Chichester, UK, 1988.

25.
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.

26.
C. Johnson, Discontinuous Galerkin finite element methods for second order hyperbolic problems, Comput. Methods Appl. Mech. Engrg. 107 (1993), no. 1-2, 177-129.

27.
C. Johnson, Y. Nie, and V. Thomee, An a posteriori error estimate and adaptive timestep control for a backward Euler discretization of a parabolic problem, SIAM J. Numer. Anal. 27 (1990), no. 2, 277-291.

28.
G. Knowles, Finite element approximation of parabolic time optimal control problems, SIAM J. Control Optim. 20 (1982), no. 3, 414-427.

29.
O. Lakkis and C. Makridakis, Elliptic reconstruction and a posteriori error estimates for fully discrete linear parabolic problems, Math. Comp. 75 (2006), no. 256, 1627-1658.

30.
R. Li,W. Liu, H.Ma, and T. Tang, Adaptive finite element approximation for distributed elliptic optimal control problems, SIAM J. Control Optim. 41 (2002), no. 5, 1321-1349.

31.
J. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer, Berlin, 1971.

32.
J. Lions and E. Magenes, Non homogeneous boundary value problems and applications, Grandlehre B. 181, Springer-Verlag, 1972.

33.
W. Liu, H. Ma, T. Tang, and N. Yan, A posteriori error estimates for discontinuous Galerkin time-stepping method for optimal control problems governed by parabolic equations, SIAM J. Numer. Anal. 42 (2004), no. 3, 1032-1061.

34.
W. Liu and N. Yan, A posteriori error estimates for convex boundary control problems, SIAM J. Numer. Anal. 39 (2001), no. 1, 73-99.

35.
W. Liu and N. Yan, A posteriori error estimates for control problems governed by Stokes equations, SIAM J. Numer. Anal. 40 (2002), no. 5, 1850-1869.

36.
C. Makridakis and R. H. Nochetto, Elliptic reconstruction and a posteriori error estimates for parabolic problems, SIAM J. Numer. Anal. 41 (2003), no. 4, 1585-1594.

37.
C. Makridakis and R. H. Nochetto, A posteriori error analysis for higher order dissipative methods for evolution problems, Numer. Math. 104 (2006), no. 4, 489-514.

38.
R. Mcknight and W. Bosarge, The Ritz-Galerkin procedure for parabolic control problems, SIAM J. Control Optim. 11 (1973), 510-524.

39.
P. Neittaanmaki and D. Tiba, Optimal Control of Nonlinear Parabolic Systems: Theory, Algorithms and Applications, M. Dekker, New York, 1994.

40.
R. H. Nochetto, G. Savare, and C. Verdi, A posteriori error estimates for variable time step discretizations of nonlinear evolution equations, Comm. Pure Appl. Math. 53 (2000), no. 5, 525-589.