Skip to main content
SpringerLink
Log in

A priori error estimates of finite volume element method for hyperbolic optimal control problems

  • Articles
  • Progress of Projects Supported by NSFC
  • Published:
Science China Mathematics Aims and scope Submit manuscript

Abstract

In this paper, optimize-then-discretize, variational discretization and the finite volume method are applied to solve the distributed optimal control problems governed by a second order hyperbolic equation. A semi-discrete optimal system is obtained. We prove the existence and uniqueness of the solution to the semidiscrete optimal system and obtain the optimal order error estimates in L ∞(J;L 2)- and L ∞(J;H 1)-norm. Numerical experiments are presented to test these theoretical results.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Adams R A. Sobolev Spaces. New York: Academic Press, 1975

    MATH  Google Scholar 

  2. Alt W, Machenroth U. Convergence of finite element approximation to state constrained convex parabolic boundary control problems. SIAM J Control Optim, 1989, 27: 718–736

    Article  MathSciNet  MATH  Google Scholar 

  3. Brenner S, Scott R. The Mathematical Theory of Finite Element Methods. New York: Springer, 2002

    Book  MATH  Google Scholar 

  4. Cai Z. On the finite volume element method. Numer Math, 1991, 58: 713–735

    Article  MathSciNet  MATH  Google Scholar 

  5. Chatzipantelidis P. A finite volume method based on the Crouzeix-Raviart element for elliptic PDEs in two dimensions. Numer Math, 1999, 82: 409–432

    Article  MathSciNet  MATH  Google Scholar 

  6. Chatzipantelidis P, Lazarov R, Thomée V. Error estimate for a finite volume element method for parabolic equations in convex polygonal domains. Numer Methods Partial Differential Equations, 2004, 20: 650–674

    Article  MathSciNet  MATH  Google Scholar 

  7. Chen L. A new class of high order finite volume methods for second order elliptic equations. SIAM J Numer Anal, 2010, 47: 4021–4043

    Article  MathSciNet  MATH  Google Scholar 

  8. Chen Y. Superconvergence of mixed finite element methods for optimal control problems. Math Comp, 2008, 77: 1269–1291

    Article  MathSciNet  MATH  Google Scholar 

  9. Chen Y, Huang Y, Liu W B, et al. Error estimates and superconvergence of mixed finite element methods for convex optimal control problems. J Sci Comput, 2010, 42: 382–403

    Article  MathSciNet  MATH  Google Scholar 

  10. Chen Y, Huang Y, Yi N. A posteriori error estimates of spectral method for optimal control problems governed by parabolic equations. Sci China Ser A, 2008, 51: 1376–1390

    Article  MathSciNet  MATH  Google Scholar 

  11. Chen Y, Liu W B. Error estimates and superconvergence of mixed finite element for quadratic optimal control. Int J Numer Anal Model, 2006, 3: 311–321

    MathSciNet  MATH  Google Scholar 

  12. Chen Y, Lu Z. Error estimates of fully discrete mixed finite element methods for semilinear quadratic parabolic optimal control problems. Comput Methods Appl Mech Engrg, 2010, 199: 1415–1423

    Article  MathSciNet  MATH  Google Scholar 

  13. Chen Y, Yi N, Liu W B. A Legendre Galerkin spectral method for optimal control problems governed by elliptic equations. SIAM J Numer Anal, 2008, 46: 2254–2275

    Article  MathSciNet  MATH  Google Scholar 

  14. Chen Z, Li R, Zhou A. A note on the optimal L 2 estimate of the finite volume element method. Adv Comput Math, 2002, 16: 291–303

    Article  MathSciNet  MATH  Google Scholar 

  15. Chou S H, Li Q. Error estimates in L 2, H 1 and L ∞ in covolume methods for elliptic and parabolic problems: A unified approach. Math Comp, 2000, 69: 103–120

    Article  MathSciNet  MATH  Google Scholar 

  16. Chou S H, Ye X. Unified analysis of finite volume methods for second order elliptic problems. SIAM J Numer Anal, 2007, 45: 1639–1653

    Article  MathSciNet  MATH  Google Scholar 

  17. Collis S, Heinkenschloss M. Analysis of the streamline upwind/Petrov Galerkin method applied to the solution of optimal control problems. CAAM Technical Report 02-01, 2002

  18. Du Q, Ju L. Finite volume methods on spheres and spherical centroidal voronoi meshes. SIAM J Numer Anal, 2005, 43: 1673–1692

    Article  MathSciNet  MATH  Google Scholar 

  19. Ewing R E, Lin T, Lin Y. On the accuracy of the finite volume element method based on piecewise linear polynomials. SIAM J Numer Anal, 2002, 39: 1865–1888

    Article  MathSciNet  MATH  Google Scholar 

  20. Gong W, Yan N. A posteriori error estimates for boundary control problems governed by parabolic partial differential equations. J Comp Math, 2009, 27: 68–88

    MathSciNet  MATH  Google Scholar 

  21. Hinze M. A variational discretization concept in control constrained optimization: The linear-quadratic case. Comput Optim Appl, 2005, 30: 45–61

    Article  MathSciNet  MATH  Google Scholar 

  22. Knowles G. Finite element approximation of parabolic time optimal control problems. SIAM J Control Optim, 1982, 20: 414–427

    Article  MathSciNet  MATH  Google Scholar 

  23. Kröner A. Adaptive finite element methods for optimal control of second order hrperbolic equations. Comput Methods Appl Math, 2011, 11: 214–240

    MathSciNet  Google Scholar 

  24. Kröner A, Kunisch K, Vexler B. Semismooth Newton methods for optimal control of the wave equation with control constraints. SIAM J Control Optim, 2011, 49: 830–858

    Article  MathSciNet  MATH  Google Scholar 

  25. Kumar S, Nataraj N, Pani A K. Finite volume element method for second order hyperbolic equations. Int J Numer Anal Model, 2008, 5: 132–151

    MathSciNet  MATH  Google Scholar 

  26. Lions J L. Optimal Control of Systems Governed by Partial Differential Equations. Berlin: Springer-Verlag, 1971

    Book  MATH  Google Scholar 

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

    Google Scholar 

  28. Luo X, Chen Y, Huang Y, et al. Some error estimates of finite volume element method for parabolic optimal control problems. Optim Contr Appl Meth, 2012, doi: 10.1002/oca.2059

  29. Luo X, Chen Y, Huang Y. A priori error analysis of Crank-Nicolson finite volume element method for parabolic optimal control problems. 2012, submitted

  30. Mcknight R, Borsarge Jr. The Ritz-Galerkin procedure for parabolic control problems. SIAM J Control Optim, 1973, 11: 510–524

    Article  MATH  Google Scholar 

  31. Meidner D, Vexler B. A priori error etimates for space-time finite element discretization of parabolic control problems, I: Problems with control constraints. SIAM J Control Optim, 2008, 47: 1301–1329

    Article  MathSciNet  MATH  Google Scholar 

  32. Meyer C, Rösch A. Superconvergence properties of optimal control problems. SIAM J Control Optim, 2004, 43: 970–985

    Article  MathSciNet  MATH  Google Scholar 

  33. Neittaanmaki P, Tiba D. Optimal Control of Nonlinear Parabolic Systems: Theory, Algorithms and Applications. New York: Marcel Dekker, 1994

    Google Scholar 

  34. Rincon A, Liu I S. On numerical approximation of an optimal control problem in linear elasticity. Divulg Mat, 2003, 11: 91–107

    MathSciNet  MATH  Google Scholar 

  35. Rösch A. Error estimates for parabolic optimal control problems with control constraints. Z Anal Anwend, 2004, 23: 353–376

    Article  MATH  Google Scholar 

  36. Tröltzsch F. Semidiscrete Ritz-Galerkin approximation of nonlinear parabolic boundary control problems-strong convergence of optimal control. Appl Math Optim, 1994, 29: 309–329

    Article  MathSciNet  MATH  Google Scholar 

  37. Xing X, Chen Y, Yi N. Error estimates of mixed finite element methods for optimal control problems governed by parabolic equations. Int J Numer Meth Eng, 2008, 5: 441–456

    MATH  Google Scholar 

  38. Yan N, Zhou Z. A priori and a posteriori error estimates of streamline diffusion finite element method for optimal control problem governed by convection dominated diffusion equation. Numer Math Theor Meth Appl, 2008, 1: 297–320

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Computational Science, Xiangtan University, Xiangtan, 411105, China

    XianBing Luo & YunQing Huang

  2. School of Science, Guizhou University, Guiyang, 550025, China

    XianBing Luo

  3. School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, China

    YanPing Chen

Authors
  1. XianBing Luo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. YanPing Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. YunQing Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to YanPing Chen.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Luo, X., Chen, Y. & Huang, Y. A priori error estimates of finite volume element method for hyperbolic optimal control problems. Sci. China Math. 56, 901–914 (2013). https://doi.org/10.1007/s11425-013-4573-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-013-4573-5

Keywords

MSC(2010)

Search

Navigation