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Computational Optimization and Applications

, Volume 71, Issue 1, pp 273–306 | Cite as

A non-overlapping DDM combined with the characteristic method for optimal control problems governed by convection–diffusion equations

  • Tongjun Sun
  • Keying Ma
Article
  • 63 Downloads

Abstract

In this paper, we consider a non-overlapping domain decomposition method combined with the characteristic method for solving optimal control problems governed by linear convection–diffusion equations. The whole domain is divided into non-overlapping subdomains, and the global optimal control problem is decomposed into the local problems in these subdomains. The integral mean method is utilized for the diffusion term to present an explicit flux calculation on the inter-domain boundary in order to communicate the local problems on the interfaces between subdomains. The convection term is discretized along the characteristic direction. We establish the fully parallel and discrete schemes for solving these local problems. A priori error estimates in \(L^2\)-norm are derived for the state, co-state and control variables. Finally, we present numerical experiments to show the validity of the schemes and verify the derived theoretical results.

Keywords

Convection–diffusion equations Optimal control problems Non-overlapping DDM Integral mean method Error estimates 

References

  1. 1.
    Peaceman, D.W.: Fundamentals of Numerical Reservoir Simulation. Elsevier, Amsterdam (1977)Google Scholar
  2. 2.
    Grossmann, C., Roos, H.G., Stynes, M.: Numerical Treatment of Partial Differential Equations. Springer, Berlin (2007)CrossRefzbMATHGoogle Scholar
  3. 3.
    Lions, J.L.: Optimal Control of Systems Governed by Partial Differential Equations. Springer, Berlin (1971)CrossRefzbMATHGoogle Scholar
  4. 4.
    Neittaanmäki, P., Tiba, D.: Optimal Control of Nonlinear Parabolic Systems, Theroy, Algorithms and Applications. Marcel Dekker, INC., New York (1994)zbMATHGoogle Scholar
  5. 5.
    Liu, W.B., Yan, N.N.: Adaptive Finite Element Method for Optimal Control Governed by PDEs. Science Press, Beijing (2008)Google Scholar
  6. 6.
    Ge, L., Liu, W.B., Yang, D.P.: Adaptive finite element approximation for a constrained optimal control problem via multi-meshes. J. Sci. Comput. 41(2), 238–255 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Yan, N.N., Zhou, Z.J.: A prior and a posteriori error analysis of edge stabilization Galerkin method for the optimal control problem governed by convection-dominated diffusion equation. J. Comput. Appl. Math. 223(1), 198–217 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bartlett, R.A., Heinkenschloss, M., Ridzal, D., Waanders, B.V.B.: Domain decomposition methods for advection dominated linear-quadratic elliptic optimal control problems. Comput. Methods Appl. Mech. Eng. 195, 6428–6447 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Becker, R., Vexler, B.: Optimal control of the convection–diffusion equation using stabilized finite element methods. Numer. Math. 106(3), 349–367 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Heinkenschloss, M., Leykekhman, D.: Local error estimates for SUPG solutions of advection-dominated elliptic linear-quadratic optimal control problems. SIAM J. Numer. Anal. 47(6), 4607–4638 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Zhu, J., Zeng, Q.: A mathematical theoretical frame for control of air pollution. Sci. China Ser. D 32, 864–870 (2002)Google Scholar
  12. 12.
    Bensoussan, A., Glowinski, R., Lions, J.L.: Mééthode de décomposition appliquée au contrôle optimal de systèmes distribués. In: 5th IFIP Conference on Optimization Techniques, Lecture Notes in Computer Science, p. 5. Springer, Berlin (1973)Google Scholar
  13. 13.
    Lagnese, J.E., Leugering, G.: Domain Decomposition Methods in Optimal Control of Partial Differential Equations. Volume 148 of International Series of Numerical Mathematics, p. 148. Birkhäuser, Basel (2004)CrossRefGoogle Scholar
  14. 14.
    Benamou, J.D.: Domain decomposition methods with coupled transmission conditions for the optimal control of systems governed by elliptic partial differential equations. SIAM J. Numer. Anal. 33(6), 2401–2416 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Heinkenschloss, M., Nguyen, H.: Neumann–Neumann domain decomposition preconditioners for linear-quadratic elliptic optimal control problems. SIAM J. Sci. Comput. 28(3), 1001–1028 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Heinkenschloss, M., Herty, M.: A spatial domain decomposition method for parabolic optimal control problems. J. Comput. Appl. Math. 201, 88–111 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Hou, L.S., Lee, J.: A Robin–Robin non-overlapping domain decomposition method for an elliptic boundary control problem. Int. J. Numer. Anal. Model. 8(3), 443–465 (2011)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Benamou, J.D.: Domain decomposition, optimal control of system governed by partial differential equations, and sysnthesis of feedback laws. J. Optim. Theory App. 102(1), 15–36 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Ewing, R.E., Russell, T.F., Wheeler, M.F.: Convergence analysis of an approximation of miscible displacement in porous media by mixed finite elements and a modified method of characteristics. Comput. Methods Appl. Mech. Eng. 47, 73–92 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Russell, T.F.: Time stepping along characteristics with incomplete iteration for a Galerkin approximation of miscible displacement in porous media. SIAM J. Numer. Anal 22, 970–1013 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Yuan, Y.R., Chang, L., Li, C.F., Sun, T.J.: Domain decomposition modified with characteristic finite element method for numerical simulation of semiconductor transient problem of heat conduction. J. Math. Res. 7(3), 61–74 (2015)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Huang, C.S.: The modified method of characteristics with adjusted advection and an accelerated domain decomposition procedure. Ph.D. thesis, Purdue University, ProQuest LLC, Ann Arbor, MI (1998)Google Scholar
  23. 23.
    Ma, K.Y., Sun, T.J., Yang, D.P.: Parallel Galerkin domain decomposition procedures for parabolic equation on general domain. Numer. Methods Partial Differ. Equ. 25(5), 1167–1194 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Ma, K.Y., Sun, T.J.: Galerkin domain decomposition procedures for parabolic equations on rectangular domain. Int. J. Numer. Methods Fluids 62(4), 449–472 (2010)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Sun, T.J., Ma, K.Y.: Dynamic parallel Galerkin domain decomposition procedures with grid modification for parabolic equation. Int. J. Numer. Methods Fluids 66(12), 1506–1529 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Sun, T.J., Ma, K.Y.: Domain decomposition procedures combined with \(H^1\)-Galerkin mixed finite element method for parabolic equation. J. Comput. Appl. Math. 267, 33–48 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Dawson, C.N., Dupont, T.F.: Explicit/Implicit conservative Galerkin domain decomposition procedures for parabolic problems. Math. Comput. 58, 21–35 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Sun, T.J., Ma, K.Y.: Parallel Galerkin domain decomposition procedures for wave euqation. J. Comput. Appl. Math. 233, 1850–1865 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Ma, K.Y., Sun, T.J.: Parallel Galerkin domain decomposition procedures based on the streamline diffusion method for convection-diffusion problems. J. Comput. Appl. Math. 235(15), 4464–4479 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Adams, R.: Sobolev Spaces. Academic, New York (1975)zbMATHGoogle Scholar
  31. 31.
    Lefschetz, S.: Differential Equations: Geometric Theory. Dover, New York (1979)zbMATHGoogle Scholar
  32. 32.
    Fu, H.F., Rui, H.X.: A priori error estimates for optimal control problems governed by transient advection–diffusion equations. J. Sci. Comput. 38, 290–315 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Houston, P.: Adaptive Lagrange–Galerkin methods for unsteady convection–diffusion problems. Math. Comput. 70, 77–106 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Süli, E.: Convergence and nonlinear stability of the Lagrange–Galerkin method for the Navier–Stokes equation. Numer. Math. 53, 459–483 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Douglas, J., Russell, T.F.: Numerical methods for convection-dominated diffusion problems based on combining the method of with finite element or finite difference procedures. SIAM J. Numer. Anal. 19, 871–885 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Zhu, J.L., Yuan, Z.Q.: Boundary Element Analysis. Science Press, Beijing (2009). (in Chinese) Google Scholar
  37. 37.
    Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Method. Volume 15 of Texts in Applied Mathematics. Springer, Berlin (1996)Google Scholar
  38. 38.
    Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland Publication, Amsterdam (1978)zbMATHGoogle Scholar
  39. 39.
    Nitsche, J.: \(L^{\infty }\)-error analysis for finite elements. In: Whiteman, J.R. (ed.) The Mathematics of Finite Elements and Applications III, pp. 173–186. Academic Press, New York (1979)Google Scholar
  40. 40.
    Schatz, A.H., Wahlbin, L.B.: Maximun norm estimates in the finite element method on plane polygonal domains, I. Math. Comput. 32, 73–109 (1978)zbMATHGoogle Scholar
  41. 41.
    Schatz, A.H., Wahlbin, L.B.: Maximun norm estimates in the finite element method on plane polygonal domains, II. Math. Comput. 33, 485–492 (1979)Google Scholar

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Authors and Affiliations

  1. 1.School of MathematicsShandong UniversityJinanPeople’s Republic of China

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