Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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ERROR ESTIMATES OF RT1 MIXED METHODS FOR DISTRIBUTED OPTIMAL CONTROL PROBLEMS

  • Hou, Tianliang (Key Laboratory for Nonlinear Science and System Structure School of Mathematics and Statistics Chongqing Three Gorges University)
  • Received : 2012.12.24
  • Published : 2014.01.31
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Abstract

In this paper, we investigate the error estimates of a quadratic elliptic control problem with pointwise control constraints. The state and the co-state variables are approximated by the order k = 1 Raviart-Thomas mixed finite element and the control variable is discretized by piecewise linear but discontinuous functions. Approximations of order $h^{\frac{3}{2}}$ in the $L^2$-norm and order h in the $L^{\infty}$-norm for the control variable are proved.

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References

  1. 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. https://doi.org/10.1023/A:1020576801966
  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), no. 1, 113-132. https://doi.org/10.1137/S0363012999351097
  3. F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer-Verlag, New York, 1991.
  4. Y. Chen, Superconvergence of mixed finite element methods for optimal control problems, Math. Comp. 77 (2008), no. 263, 1269-1291. https://doi.org/10.1090/S0025-5718-08-02104-2
  5. Y. Chen, Superconvergence of quadratic optimal control problems by triangular mixed finite elements, Internat. J. Numer. Methods Engrg. 75 (2008), no. 8, 881-898. https://doi.org/10.1002/nme.2272
  6. Y. Chen and Y. Dai, Superconvergence for optimal control problems governed by semilinear elliptic equations, J. Sci. Comput. 39 (2009), no. 2, 206-221.
  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), no. 3, 382-403.
  8. 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. https://doi.org/10.1016/j.cam.2006.11.015
  9. P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978.
  10. P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, Boston-London-Melbourne, 1985.
  11. J. Douglas and J. E. Roberts, Global estimates for mixed methods for second order elliptic equations, Math. Comp. 44 (1985), no. 169, 39-52. https://doi.org/10.1090/S0025-5718-1985-0771029-9
  12. F. S. Falk, Approximation of a class of optimal control problems with order of convergence estimates, J. Math. Anal. Appl. 44 (1973), 28-47. https://doi.org/10.1016/0022-247X(73)90022-X
  13. T. Geveci, On the approximation of the solution of an optimal control problem governed by an elliptic equation, RAIRO. Anal. Numer. 13 (1979), no. 4, 313-328. https://doi.org/10.1051/m2an/1979130403131
  14. R. Li, W. B. Liu, H. P. Ma, and T. Tang, Adaptive finite element approximation for distributed elliptic optimal control problems, SIAM J. Control Optim. 41 (2002), no. 5, 1321-1349. https://doi.org/10.1137/S0363012901389342
  15. R. Li and W. Liu, http://circus.math.pku.edu.cn/AFEPack.
  16. J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, Berlin, 1971.
  17. W. B. Liu and N. N. Yan, A posteriori error analysis for convex distributed optimal control problems, Adv. Comp. Math. 15 (2001), 285-309. https://doi.org/10.1023/A:1014239012739
  18. Z. Lu and Y. Chen, $L^{\infty}$-error estimates of triangular mixed finite element methods for optimal control problems governed by semilinear elliptic equations, Numer. Anal. Appl. 2 (2009), no. 1, 74-86. https://doi.org/10.1134/S1995423909010078
  19. C. Meyer and A. Rosch, Superconvergence properties of optimal control problems, SIAM J. Control Optim. 43 (2004), no. 3, 970-985. https://doi.org/10.1137/S0363012903431608
  20. C. Meyer and A. Rosch, $L^{\infty}$-error estimates for approximated optimal control problems, SIAM J. Control Optim. 44 (2005), 1636-1649. https://doi.org/10.1137/040614621
  21. P. A. Raviart and J. M. Thomas, A mixed finite element method for 2nd order elliptic problems, aspects of the finite element method, Lecture Notes in Math, Springer, Berlin, 606 (1977), 292-315.
  22. X. Xing and Y. Chen, $L^{\infty}$-error estimates for general optimal control problem by mixed finite element methods, Int. J. Numer. Anal. Model. 5 (2008), no. 3, 441-456.