Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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A MULTIGRID METHOD FOR AN OPTIMAL CONTROL PROBLEM OF A DIFFUSION-CONVECTION EQUATION

  • Baek, Hun-Ki (Department of Mathematics Kyungpook National University) ;
  • Kim, Sang-Dong (Department of Mathematics Kyungpook National University) ;
  • Lee, Hyung-Chun (Department of Mathematics Ajou University)
  • Published : 2010.01.01
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Abstract

In this article, an optimal control problem associated with convection-diffusion equation is considered. Using Lagrange multiplier, the optimality system is obtained. The derived optimal system becomes coupled, non-symmetric partial differential equations. For discretizations and implementations, the finite element multigrid V-cycle is employed. The convergence analysis of finite element multigrid methods for the derived optimal system is shown. Some numerical simulations are performed.

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References

  1. Z.-Z. Bai and M. K. Ng, On inexact preconditioners for nonsymmetric matrices, SIAM J. Sci. Comput. 26 (2005), no. 5, 1710-1724 https://doi.org/10.1137/040604091
  2. R. Becker and B. Vexler, Optimal control of the convection-diffusion equation using stabilized finite element methods, Numer. Math. 106 (2007), no. 3, 349-367 https://doi.org/10.1007/s00211-007-0067-0
  3. P. B. Bochev and M. D. Gunzburger, Least-squares finite element methods for optimality systems arising in optimization and control problems, SIAM J. Numer. Anal. 43 (2006), no. 6, 2517-2543 https://doi.org/10.1137/040607848
  4. A. Borzi, Multigrid methods for parabolic distributed optimal control problems, J. Comput. Appl. Math. 157 (2003), no. 2, 365-382 https://doi.org/10.1016/S0377-0427(03)00417-5
  5. A. Borzi and K. Kunisch, The numerical solution of the steady state solid fuel ignition model and its optimal control, SIAM J. Sci. Comput. 22 (2000), no. 1, 263-284 https://doi.org/10.1137/S1064827599360194
  6. A. Borzi, K. Kunisch, and D. Y. Kwak, Accuracy and convergence properties of the finite difference multigrid solution of an optimal control optimality system, SIAM J. Control Optim. 41 (2002), no. 5, 1477-1497 https://doi.org/10.1137/S0363012901393432
  7. J. H. Bramble, Multigrid Methods, Pitman Res. Notes Math, Ser. 294, John Wiley, New York, 1993
  8. J. H. Bramble, D. Y. Kwak, and J. E. Pasciak, Uniform convergence of multigrid V - cycle iterations for indefinite and nonsymmetric problems, SIAM J. Numer. Anal. 31 (1994), no. 6, 1746-1763 https://doi.org/10.1137/0731089
  9. J. H. Bramble and J. E. Pasciak, New estimates for multilevel algorithms including the V -cycle, Math. Comp. 60 (1993), no. 202, 447-471 https://doi.org/10.2307/2153097
  10. J. H. Bramble, J. E. Pasciak, and J. Xu, The analysis of multigrid algorithms for nonsymmetric and indefinite elliptic problems, Math. Comp. 51 (1988), no. 184, 389-414 https://doi.org/10.2307/2008755
  11. J. H. Bramble, J. E. Pasciak, and J. Xu, The analysis of multigrid algorithms with nonnested spaces or noninherited quadratic forms, Math. Comp. 56 (1991), no. 193, 1-34 https://doi.org/10.2307/2008527
  12. S. C. Brenner, Convergence of the multigrid V -cycle algorithm for second-order boundary value problems without full elliptic regularity, Math. Comp. 71 (2002), no. 238, 507-525 https://doi.org/10.1090/S0025-5718-01-01361-8
  13. S. C. Brenner and L. R. Scott, The mathematical Theory of Finite Element Methods, Springer-Verlag, New York, 2002
  14. L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 1998
  15. J. L. Guermond, Stabilization of Galerkin approximations of transport equations by subgrid modeling, M2AN Math. Model. Numer. Anal. 33 (1999), no. 6, 1293-1316 https://doi.org/10.1051/m2an:1999145
  16. M. D. Gunzburger, Perspectives in Flow Control and Optimization, Advances in Design and Control, 5. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2003
  17. M. D. Gunzburger and H.-C. Lee, Analysis, approximation, and computation of a coupled solid/fluid temperature control problem, Comput. Methods Appl. Mech. Engrg. 118 (1994), no. 1-2, 133-152 https://doi.org/10.1016/0045-7825(94)00022-0
  18. M. D. Gunzburger and H.-C. Lee, Analysis and approximation of optimal control problems for first-order elliptic systems in three dimensions, Appl. Math. Comput. 100 (1999), no. 1, 49-70 https://doi.org/10.1016/S0096-3003(98)00017-4
  19. W. Hackbusch, Multigrid Methods and Applications, Springer Series in Computational Mathematics, 4. Springer-Verlag, Berlin, 1985
  20. J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, Berlin, 1971
  21. P. Neittaanmaki and D. Tiba, Optimal Control of Nonlinear Parabolic Systems, Marcel Dekker, New York, 1994
  22. M. H. van Raalte and P. W. Hemker, Two-level multigrid analysis for the convectiondiffusion equation discretized by a discontinuous Galerkin method, Numer. Linear Algebra Appl. 12 (2005), no. 5-6, 563-584 https://doi.org/10.1002/nla.441
  23. A. H. Schatz, An observation concerning Ritz-Galerkin methods with indefinite bilinear forms, Math. Comp. 28 (1974), 959-962 https://doi.org/10.2307/2005357
  24. J. Wang, Convergence analysis of multigrid algorithms for nonselfadjoint and indefinite elliptic problems, SIAM J. Numer. Anal. 30 (1993), no. 1, 275-285 https://doi.org/10.1137/0730013

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