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DOI QR Code

FINITE ELEMENT APPROXIMATIONS OF THE OPTIMAL CONTROL PROBLEMS FOR STOCHASTIC STOKES EQUATIONS

  • Choi, Youngmi (College of Liberal Arts Anyang University) ;
  • Kim, Soohyun (Department of Mathematics Ajou University) ;
  • Lee, Hyung-Chun (Department of Mathematics Ajou University)
  • Received : 2013.05.12
  • Published : 2014.05.31

Abstract

Finite element approximation solutions of the optimal control problems for stochastic Stokes equations with the forcing term perturbed by white noise are considered. Error estimates are established for the fully coupled optimality system using Brezzi-Rappaz-Raviart theory. Numerical examples are also presented to examine our theoretical results.

Keywords

stochastic Stokes equations;optimal control;white noise

References

  1. F. Abergal and R. Temmam, On some control problems in fluid mechanics, Theoret. Comput. Fluid Dynamics 1 (1990), 303-325. https://doi.org/10.1007/BF00271794
  2. R. Temam, Nonlinear Functional Analysis and Navier-Stokes Equations, SIAM, Philadelphia, 1983.
  3. L. S. Hou, J. Lee, and H. Manouzi, Finite element approximations of stochastic optimal control problems constrained by stochastic elliptic PDEs, J. Math. Anal. Appl. 384 (2011), no. 1, 87-103. https://doi.org/10.1016/j.jmaa.2010.07.036
  4. M. D. Gunzburger, L. S. Hou, and T. P. Svobodny, Analysis and finite element approximation of optimal control problems for the stationary Navier-Stokes equations with distributed and Neumann controls, Math. Comp. 57 (1991), no. 195, 123-151. https://doi.org/10.1090/S0025-5718-1991-1079020-5
  5. M. Gunzburger, H.-C. Lee, and J. Lee, Error estimates of stochastic optimal neumann boundary control problems, SIAM J. Numer. Anal. 49 (2011), no. 4, 1532-1552. https://doi.org/10.1137/100801731
  6. M. D. Gunzburger and S. Manservisi, Analysis and approximation of the velocity tracking problem for Navier-Stokes flows with distributed control, SIAM J. Numer. Anal. 37 (2000), no. 5, 1481-1512. https://doi.org/10.1137/S0036142997329414
  7. H. -C. Lee, Analysis and computational methods of Dirichlet boundary optimal control problems for 2D Boussinesq equations, Adv. Comput. Math. 19 (2003), no. 1-3, 255-275. https://doi.org/10.1023/A:1022872602498
  8. H.-C. Lee and Y. Choi, A least-squares method for optimal control problems for a secondorder elliptic systems in two dimensions, J. Math. Anal. Appl. 242 (2000), no. 1, 105-128. https://doi.org/10.1006/jmaa.1999.6658
  9. H.-C. Lee and O. Y. Imanuvilov, Analysis pf optimal control problems for 2D stationary Boussinesq equations, J. Math. Anal. Appl. 242 (2000), 191-211. https://doi.org/10.1006/jmaa.1999.6651
  10. H.-C. Lee and S. Kim, Finite element approximation and computations of optimal dirichlet boundary control problems for the boussinesq equations, J. Korean Math. Soc. 41 (2004), no. 4, 681-715. https://doi.org/10.4134/JKMS.2004.41.4.681
  11. J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer, New York, 1971.
  12. H. Manouzi, A finite element approximation of linear stochastic PDEs driven by multiplicative white noise, Int. J. Comput. Math. 85 (2008), no. 3-4, 527-546. https://doi.org/10.1080/00207160701210133
  13. H. Manouzi and L. S. Hou, An optimal control problem for stochastic linear PDEs driven by a Gaussian white noise, 629-636, Numerical Mathematics and Advanced Applications, Springer Berlin Heidelberg, 2008.
  14. G. Stefanou, The stochastic finite element method: Past, present, and future, Comput. Methods Appl. Mech. Engrg. 198 (2009), Issues 9-12, 1031-1051. https://doi.org/10.1016/j.cma.2008.11.007
  15. R. Adams, Sobolev Spaces, Academic Press, New York, 1975.
  16. I. Babuska, R. Tempone, and G. E. Zouraris, Galerkin finite element approximations of stochastic elliptic partial differential equations, SIAM J. Numer. Anal. 42 (2004), no. 2, 800-825. https://doi.org/10.1137/S0036142902418680
  17. I. Babuska, R. Tempone, and G. E. Zouraris, Solving elliptic boundary value problems with uncertain coecients by the finite element method: The stochastic formulation, Comput. Methods Appl. Mech. Engrg. 194 (2005), no. 12-16, 1251-1294. https://doi.org/10.1016/j.cma.2004.02.026
  18. F. Brezzi, J. Rappaz, and P. Raviart, Finite-dimensional approximation of nonlinear problems. I. Branches of nonsingular solutions, Numer. Math. 36 (1980), 1-25. https://doi.org/10.1007/BF01395985
  19. Y. Cao, Z. Chen, and M. Gunzburger, Error analysis of finite element approximations of the stochastic Stokes equations, Adv. Comput. Math. 33 (2010), no. 2, 215-230. https://doi.org/10.1007/s10444-009-9127-6
  20. Y. Cao, H. Yang, and L. Yin, Finite element methods for semilinear elliptic stochastic partial differential equations, Numer. Math. 106 (2007), no. 2, 181-198. https://doi.org/10.1007/s00211-007-0062-5
  21. P. Ciarlet, Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978.
  22. I. Babuska and P. Chatzipantelidis, On solving elliptic stochastic partial differential equations, Comput. Methods Appl. Mech. Engrg. 191 (2002), no. 37-38, 4093-4122. https://doi.org/10.1016/S0045-7825(02)00354-7
  23. P. Ciarlet, Introduction to Numerical Linear Algebra and Optimization, Cambridge, 1989.
  24. Y. Choi, H.-C. Lee, and S. D. Kim, Analysis and computations of least-squares method for optimal control problems for the Stokes equations, J. Korean Math. Soc. 46 (2009), no. 5, 1007-1025. https://doi.org/10.4134/JKMS.2009.46.5.1007
  25. Y. Choi, H.-C. Lee, and B.-C. Shin, A least-square/penalty method for distributed optimal control problems for Stokes equations, Comput. Math. Appl. 53 (2007), no. 11, 1672-1685. https://doi.org/10.1016/j.camwa.2007.01.009
  26. R. G. Ghanem and P. D. Spanos, Stochasic Finite Elements: A spectral approach, Springer-Verlag, 1991.
  27. V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations, Springer, Berlin, 1986.
  28. M. D. Gunzburger and L. S. Hou, Finite-dimensional approximation of a class of constrained nonlinear optimal control problems, SIAM J. Control. Optim. 34 (1996), no. 3, 1001-1043. https://doi.org/10.1137/S0363012994262361