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A RT Mixed FEM/DG Scheme for Optimal Control Governed by Convection Diffusion Equations

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Abstract

In this paper, we provide a numerical scheme—RT mixed FEM/DG scheme for the constrained optimal control problem governed by convection dominated diffusion equations. A priori and a posteriori error estimates are obtained for both the state, the co-state and the control. The adaptive mesh refinement can be applied indicated by a posteriori error estimator provided in this paper. Numerical examples are presented to illustrate the theoretical analysis.

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References

  1. Bartlett, R., Heinkenschloss, M., Ridzal, D., Van Bloemen Waanders, B.: Domain decomposition methods for advection dominated linear quadratic elliptic optimal control problems. Technical Report SAND 2005-2895, Sandia National Laboratories (2005)

  2. Becker, R., Vexler, B.: Optimal control of the convection-diffusion equation using stabilized finite element methods. Numer. Math. 106(3), 349–367 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  3. Behr, M., Heinkenschloss, M.: The effect of stabilization in finite element methods for the optimal boundary control of the Oseen equations. Finite Elem. Anal. Des. 41, 229–251 (2004)

    Article  MathSciNet  Google Scholar 

  4. Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer, Berlin (1991)

    MATH  Google Scholar 

  5. Brezzi, F., Russo, A.: Choosing bubbles for advection-diffusion problems. Math. Models Methods Appl. Sci. 4, 571–587 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  6. France, L.P., Frey, S.L., Hughes, T.J.R.: Stabilized finite element methods I: Application to the convection diffusive model. Comput. Methods Appl. Mech. Eng. 95, 253–276 (1992)

    Article  Google Scholar 

  7. Fursikov, A.V.: Optimal Control of Distributed Systems, Theory and Applications. American Mathematical Society, Providence (2000)

    Google Scholar 

  8. Huang, Y.Q., Li, R., Liu, W.B., Yan, N.N.: Efficient discretization to finite element approximation of constrained optimal control problems. SIAM J. Control Optim. (to appear)

  9. Hughes, T.J.R., Brooks, A.: Streamline upwind/Petrov Galerkin formulations for the convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations. Comput. Methods Appl. Mech. Eng. 54, 199–259 (1982)

    MathSciNet  Google Scholar 

  10. Johnson, C.: Numerical Solution of Partial Differential Equations by the Finite Element Method. Cambridge Univ. Press, Cambridge (1987)

    MATH  Google Scholar 

  11. Johnson, C., Pitkränta, J.: An analysis of the discontinuous Galerkin method for scalar hyperbolic equation. Math. Comput. 46, 1–26 (1986)

    Article  MATH  Google Scholar 

  12. Kim, D., Park, E.: A posteriori error estimates for the upstream weighting mixed method for convection diffusion problems. Comput. Methods Appl. Mech. Eng. 197, 806–820 (2008)

    Article  MATH  Google Scholar 

  13. Lions, J.L.: Optimal Control of Systems Governed by Partial Differential Equations. Springer, Berlin (1971)

    MATH  Google Scholar 

  14. Liu, W.B., Yan, N.N.: Adaptive Finite Element Methods for Optimal Control Governed by PDEs. Science Press, Beijing (2008)

    Google Scholar 

  15. Navert, U.: A finite element method for convection diffusion problems. Ph.D. thesis, Chalmers Inst. of Tech. (1982)

  16. Raviart, P.A., Thomas, J.M.: A mixed finite element method for 2nd order elliptic problems. In: Math. Aspects of the Finite Element Method. Lecture Notes in Math., vol. 606, pp. 292–315. Springer, Berlin (1977)

    Chapter  Google Scholar 

  17. Scott Collis, S., Heinkenschloss, M.: Analysis of the streamline upwind/Petrov Galerkin method applied to the solution of optimal control problems. CAAM TR02-01 (March 2002)

  18. Wang, J.P., Yan, N.N.: A parallel domain decomposition procedure for convection diffusion problems. In: Glowinski, R., Periaux, J., Shi, Z.C., Widlund, O. (eds.) Domain Decomposition Methods in Science and Engineerings, pp. 331–339. Wiley, Chichester (1997)

    Google Scholar 

  19. Zhu, J., Zeng, Q.C.: A mathematical theoretical frame for control of air pollution. Sci. China, Ser. D 32, 864–870 (2002)

    Google Scholar 

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Authors and Affiliations

  1. LSEC, Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Science, Beijing, 100080, China

    Ningning Yan & Zhaojie Zhou

Authors
  1. Ningning Yan
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  2. Zhaojie Zhou
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Corresponding author

Correspondence to Zhaojie Zhou.

Additional information

The research was supported by the National Basic Research Program under the Grant 2005CB321701 and the National Natural Science Foundation of China under the Grant 10771211.

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Yan, N., Zhou, Z. A RT Mixed FEM/DG Scheme for Optimal Control Governed by Convection Diffusion Equations. J Sci Comput 41, 273 (2009). https://doi.org/10.1007/s10915-009-9297-x

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  • DOI: https://doi.org/10.1007/s10915-009-9297-x

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