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.
Similar content being viewed by others
References
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)
Becker, R., Vexler, B.: Optimal control of the convection-diffusion equation using stabilized finite element methods. Numer. Math. 106(3), 349–367 (2007)
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)
Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer, Berlin (1991)
Brezzi, F., Russo, A.: Choosing bubbles for advection-diffusion problems. Math. Models Methods Appl. Sci. 4, 571–587 (1994)
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)
Fursikov, A.V.: Optimal Control of Distributed Systems, Theory and Applications. American Mathematical Society, Providence (2000)
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)
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)
Johnson, C.: Numerical Solution of Partial Differential Equations by the Finite Element Method. Cambridge Univ. Press, Cambridge (1987)
Johnson, C., Pitkränta, J.: An analysis of the discontinuous Galerkin method for scalar hyperbolic equation. Math. Comput. 46, 1–26 (1986)
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)
Lions, J.L.: Optimal Control of Systems Governed by Partial Differential Equations. Springer, Berlin (1971)
Liu, W.B., Yan, N.N.: Adaptive Finite Element Methods for Optimal Control Governed by PDEs. Science Press, Beijing (2008)
Navert, U.: A finite element method for convection diffusion problems. Ph.D. thesis, Chalmers Inst. of Tech. (1982)
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)
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)
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)
Zhu, J., Zeng, Q.C.: A mathematical theoretical frame for control of air pollution. Sci. China, Ser. D 32, 864–870 (2002)
Author information
Authors and Affiliations
Corresponding author
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.
Rights and permissions
About this article
Cite this article
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
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10915-009-9297-x