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ERROR ESTIMATES OF MIXED FINITE ELEMENT APPROXIMATIONS FOR A CLASS OF FOURTH ORDER ELLIPTIC CONTROL PROBLEMS

  • Hou, Tianliang (School of Mathematical Sciences South China Normal University)
  • Received : 2012.05.23
  • Published : 2013.07.31

Abstract

In this paper, we consider the error estimates of the numerical solutions of a class of fourth order linear-quadratic elliptic optimal control problems by using mixed finite element methods. The state and co-state are approximated by the order $k$ Raviart-Thomas mixed finite element spaces and the control variable is approximated by piecewise polynomials of order $k(k{\geq}1)$. $L^2$ and $L^{\infty}$-error estimates are derived for both the control and the state approximations. These results are seemed to be new in the literature of the mixed finite element methods for fourth order elliptic control problems.

Keywords

References

  1. F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer-Verlag, New York, 1991.
  2. H. Bium and R. Rannacher, On mixed finite element methods in plate bending analysis, Comput. Mech. 6 (1990), 221-236. https://doi.org/10.1007/BF00350239
  3. W. Cao and D. Yang, Ciarlet-Raviart mixed finite element approximation for an optimal control problem governed by the first bi-harmonic equation, J. Comput. Appl. Math. 233 (2009), no. 2, 372-388. https://doi.org/10.1016/j.cam.2009.07.039
  4. H. Chen and Z. Jiang, $L^{\infty}$-convergence of mixed finite element method for laplacian operator, Korean J. Comput. Appl. Math. 7 (2000), no. 1, 61-82.
  5. 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
  6. 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
  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. Comput. Appl. Math. 211 (2008), no. 1, 76-89. https://doi.org/10.1016/j.cam.2006.11.015
  9. X. L. Cheng, W. M. Han, and H. C. Huang, Some mixed finite element methods for biharmonic equation, J. Comput. Appl. Math. 126 (2000), no. 1-2, 91-109. https://doi.org/10.1016/S0377-0427(99)00342-8
  10. 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
  11. R. Duran, R. H. Nochetto, and J. Wang, Sharp maximum norm error estimates for FE approximations of the Stokes problems in 2-D, Math. Comp. 51 (1988), no. 184, 491-506.
  12. J. Frehse and R. Rannacher, Eine $L^1$-Ferhlerabschatzung fur diskrete Grundlosungen in der Methode der Finiten Elemente, Finite Elemente (Tagung, Univ. Bonn, Bonn, 1975), pp. 92-114. Bonn. Math. Schrift., No. 89, Inst. Angew. Math., Univ. Bonn, Bonn, 1976.
  13. C. Johnson, On the convergence of a mixed finite-element method for plate bending problems, Numer. Math. 21 (1973), 43-62. https://doi.org/10.1007/BF01436186
  14. B. J. Li and S. Y. Liu, On gradient-type optimization method utilizing mixed finite element approximation for optimal boundary control problem governed by bi-harmonic equation, Appl. Math. Comput. 186 (2007), no. 2, 1429-1440. https://doi.org/10.1016/j.amc.2006.07.142
  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. P. Monk, A mixed finite element method for the biharmonic equation, SIAM J. Numer. Anal. 24 (1987), no. 4, 737-749. https://doi.org/10.1137/0724048
  18. P. A. Raviart and J. M. Thomas, A mixed finite element method for 2nd order elliptic problems, Mathematical aspects of finite element methods (Proc. Conf., Consiglio Naz. delle Ricerche (C.N.R.), Rome, 1975), pp. 292-315. Lecture Notes in Math., Vol. 606, Springer, Berlin, 1977.
  19. R. Scott, A Mixed method for 4th order problems using linear finite elements, RARIO Anal. Numer. 33 (1978), 681-697.
  20. R. Scott, Optimal $L^{\infty}$-estimates for the finite element method on irregular meshes, Math. Comp. 30 (1976), no. 136, 681-697.
  21. J. Wang, Asympotic expansion and $L^{\infty}$-error estimates for mixed FEM for second order elliptic problems, Numer. Math. 55 (1989), 401-430. https://doi.org/10.1007/BF01396046
  22. X. Xing and Y. Chen, Error estimates of mixed finite element methods for quadratic optimal control problems, J. Comput. Appl. Math. 233 (2010), no. 8, 1812-1820. https://doi.org/10.1016/j.cam.2009.09.018

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