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A posteriori error estimates of spectral method for optimal control problems governed by parabolic equations

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Abstract

In this paper, we investigate the Legendre Galerkin spectral approximation of quadratic optimal control problems governed by parabolic equations. A spectral approximation scheme for the parabolic optimal control problem is presented. We obtain a posteriori error estimates of the approximated solutions for both the state and the control.

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References

  1. Ghanem R, Sissaoui H. A posteriori error estimate by a spectral method of an elliptic optimal control problem. J Comput Math Optim, 2: 111–125 (2006)

    MATH  MathSciNet  Google Scholar 

  2. Liu W B, Tiba D. Error estimates for the finite element approximation of nonlinear optimal control problems. J Numer Func Optim, 22: 953–972 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  3. Liu W B, Yan N N. A posteriori error analysis for convex distributed optimal control problems. Adv Comput. Math, 15: 285–309 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  4. Liu W B, Yan N N. A posteriori error estimates for optimal control problems governed by parabolic equations. Numer Math, 93: 497–521 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  5. Liu W B, Yan N N. A posteriori error estimates for optimal control of stokes flows. SIAM J Numer Anal, 40: 1805–1869 (2003)

    Google Scholar 

  6. Chen Y, Liu W B. A posteriori error estimates for mixed finite elements of a quadratic control problem. Recent Progress in Computational and Applied PDEs. Kluwer Academic, 2002, 123–134

  7. Chen Y, Liu W B. Error estimates and superconvergence of mixed finite element for quadratic optimal control. Int J Numer Anal Model, 3(3): 311–321 (2006)

    MATH  MathSciNet  Google Scholar 

  8. Chen Y, Liu W B. A posteriori error estimates for mixed finite element solutions of convex optimal control problems. J Comput Appl Math, 211: 76–89 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  9. Chen Y. Superconvergence of optimal control problems by rectangular mixed finite element methods. Math Comp, 77: 1269–1291 (2008)

    Article  MathSciNet  Google Scholar 

  10. Chen Y. Superconvergence of quadratic optimal control problems by triangular mixed finite elements. Int J Num Methods Engineering (in press)

  11. Xing X, Chen Y. L ∞-error estimates for general optimal control problem by mixed finite element methods. Int J Num Anal Model, 5(3): 441–456 (2008)

    Google Scholar 

  12. Xing X, Chen Y. Error estimates of mixed methods for optimal control problems governed by parabolic equations. Int J Num Methods Engineering (in press)

  13. Chen Y, Yi N, Liu W B. A Legendre Galerkin spectral method for optimal control problems governed by elliptic equations. SIAM J Numer Anal, 46(5): 2254–2275 (2008)

    Article  Google Scholar 

  14. Lions J L, Magenes E. Non Homogeneous Boundary Value Problems and Applications. Berlin: Springer-Verlag, 1972

    Google Scholar 

  15. Lions J L. Optimal Control of Systems Governed by Partial Differential Equations. Berlin: Springer-Verlag, 1971

    MATH  Google Scholar 

  16. Canuto C, Hussaini M Y, Quarteroni A, Zang T A. Spectral Methods in Fluid Dynamics. Berlin: Springer-Verlag, 1988

    MATH  Google Scholar 

  17. Guo B. Spectral Methods and Their Applications. Singapore: World Scientific, 1998

    MATH  Google Scholar 

  18. Li R, Liu W B, Ma H P, Tang T. Adaptive finite element approximation of elliptic optimal control. SIAM J Control Optim, 41(5): 1321–1349 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  19. Huang Y, R. Li, Liu W B, Yan N N. Adaptive multi-mesh finite element approximation for constrained optimal control problems. SIAM J Control Optim (in press)

  20. Liu W B, Barrett J W. Error bounds for the finite element approximation some degenerate quasilinear parabolic equations and variational inequalities. Adv Comp Math, 1(2): 223–239 (1993)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, China

    YanPing Chen

  2. Hunan Key Laboratory for Computation and Simulation in Science and Engineering, School of Mathematics and Computational Science, Xiangtan University, Xiangtan, 411105, China

    YunQing Huang & NianYu Yi

Authors
  1. YanPing Chen
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  2. YunQing Huang
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  3. NianYu Yi
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Corresponding author

Correspondence to YunQing Huang.

Additional information

The first author was supported by the National Basic Research Program, the National Natural Science Foundation of China (Grant No. 2005CB321703) and Scientific Research Fund of Hunan Provincial Education Department. The second author was supported by the Outstanding Youth Scientist of the National Natural Science Foundation of China (Grant No. 10625106), and the National Basic Research Program of China (Grant No. 2005CB321701)

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Chen, Y., Huang, Y. & Yi, N. A posteriori error estimates of spectral method for optimal control problems governed by parabolic equations. Sci. China Ser. A-Math. 51, 1376–1390 (2008). https://doi.org/10.1007/s11425-008-0097-9

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  • DOI: https://doi.org/10.1007/s11425-008-0097-9

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