Skip to main content
SpringerLink
Log in

Overlapping Domain Decomposition Method for a Noncoercive System of Quasi-Variational Inequalities Related to the Hamilton–Jacobi–Bellman Equation

  • Published:
Computational Mathematics and Modeling Aims and scope Submit manuscript

This paper deals with numerical analysis of an overlapping Schwarz method on nonmatching grids for a noncoercive system of quasi-variational inequalities related to the Hamilton–Jacobi–Bellman equation. We prove that the discrete alternating Schwarz sequences on every subdomain converge monotonically into the unique solution of the discrete problem and geometrically in uniform norm. Optimally L ∞-error estimates are also established.

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. L. Badea, “On the Schwarz alternating method with more than two subdomains for nonlinear monotone problems,” SIAM J. Numer. Anal., 28, No. 1, 179–204 (1991).

    Article  MathSciNet  MATH  Google Scholar 

  2. A. Bensoussan and J. L. Lions, “Applications des in´equations variationnelles en contrˆole stochastique,” Dunod, Paris (1978).

  3. M. Boulbrachen and M. Haiour, “On a noncoercive system of quasivariational inequalities related to stochastic control problems,” JIPAM, 30 (2002).

  4. M. Boulbrachene and M. Haiour, “The finite element approximation of Hamilton–Jacobi–Bellman equations,” Comput. Math. Appl., 41, 993–1007 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  5. P. Ciarlet and P. Raviart, “Maximum principle and uniform convergence for the finite element method,” Comput. Meth. Appl. Mech. Eng., 2, 1–20 (1973).

    Article  MathSciNet  Google Scholar 

  6. P. Cortey-Dumont, “Sur les inéquations variationnelles `a opérateur non coercif,” RAIRO, Modélisation Mathématique et Analyse Numérique, 19, No. 2, 195–212 (1985).

  7. L. C. Evans and A. Friedman, “Optimal stochastic switching and the Dirichlet Problem for the Bellman equations,” Trans. Amer. Math. Soc., 253, 365–389 (1979).

    Article  MathSciNet  MATH  Google Scholar 

  8. M. Haiour and S. Boulaaras, “Overlapping domain decomposition methods for elliptic quasi-variational inequalities related to impulse control problem with mixed boundary conditions,” Proc. Math. Sci., 121, No. 4, 481–493 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  9. P. L. Lions and J. L. Menaldi, “Optimal control of stochastic integrals and Hamilton Jacobi Bellman equations. I,” SIAM J. Control Optim., 20, 58–81 (1979).

    Article  MathSciNet  Google Scholar 

  10. S. Z. Zhou, Zhan, and G. H. Chen, “Domain decomposition methods for a system of quasi-variational inequalities,” National Natural Science Foundation of China, No. 10571046 (2008).

  11. S. Z. Zhou and G. H. Chen, “A monotone iterative algorithm for a discrete Hamilton–Jacobi–Bellman equation,” Math. Appl., 18, 639–645 (2005).

    MATH  Google Scholar 

  12. S. Z. Zhou and J. P. Zeng, “Schwarz algorithm for obstacle problems with a type of nonlinear operators,” Acta Math. Appl. Sin., 20, 521–530 (1997).

    MathSciNet  MATH  Google Scholar 

  13. J. Zeng and S. Z. Zhou, “Schwarz algorithm for the solution of variational inequalities with nonlinear source terms,” Appl. Math. Comput., 97, 23–35 (1998).

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Annaba, Annaba, Algeria

    L. Djemaoune & M. Haiour

Authors
  1. L. Djemaoune
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. Haiour
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to L. Djemaoune.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Djemaoune, L., Haiour, M. Overlapping Domain Decomposition Method for a Noncoercive System of Quasi-Variational Inequalities Related to the Hamilton–Jacobi–Bellman Equation. Comput Math Model 27, 217–227 (2016). https://doi.org/10.1007/s10598-016-9316-3

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10598-016-9316-3

Keywords

Search

Navigation