Skip to main content
SpringerLink
Log in

High order finite volume methods for singular perturbation problems

  • Published:
Science in China Series A: Mathematics Aims and scope Submit manuscript

Abstract

In this paper we establish a high order finite volume method for the fourth order singular perturbation problems. In conjunction with the optimal meshes, the numerical solutions resulting from the method have optimal convergence order. Numerical experiments are presented to verify our theoretical estimates.

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. Stynes M, O’Riordan E. A finite element method for a singularly perturbed boundary value problem. Numer Math, 50: 1–15 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  2. Shishkin G I. Grid approximation of singularly perturbed parabolic equations with internal layers. Sov J Numer Anal Math Model, 3: 393–407 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  3. Stynes M, O’Riordan E. An analysis of a singularly perturbed two-point boundary value problem using only finite element techniques. Math Comp, 56: 663–675 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  4. Sun G, Stynes M. Finite element methods for singularly perturbed high-order elliptic two-point boundary problems, I: Reaction-diffusion-type problems. IMA J Numer Anal, 15: 117–139 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  5. Sun G, Stynes M. Finite element methods for singularly perturbed high-order elliptic two-point boundary problems, II: Convection-diffusion-type problems. IMA J Numer Anal, 15: 197–219 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  6. Miller J J H, O’Riordan E, Shishkin G I. Fitted Numerical Methods for Singularly Perturbation Problems. Singapore: World Scientific, 1996

    Google Scholar 

  7. Roos H G, Stynes M, Tobiska L. Numerical Methods for Singularly Perturbed Differential Equations. Berlin: Springer-Verlag, 1996

    MATH  Google Scholar 

  8. Tang T, Trummer M R. Boundary layer resolving pseudospectral methods for singular perturbation problems. SIAM J Sci Comput, 17: 430–438 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  9. Qiu Y, Sloan D M. Analysis of difference approximations to a singular perturbed two-point boundary value problem on an adaptively generated grid. J Comput Appl Math, 101: 1–25 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  10. Liu W, Tang T. Error analysis for a Galerkin-spectral method with coordinate transformation for solving singularly perturbed problems. Appl Numer Math, 38: 315–345 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  11. Liu S, Xu Y. Galerkin methods based on Hermite splines for singular perturbation problems. SIAM J Number Anal, 43: 2607–2623 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  12. Stynes M. Steady-state convection-diffusion problems. Acta Numer, 14: 445–508 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  13. Cao Y Z, Herdman T, Xu Y. A hybrid collocation method for Volterra integral equations with weakly singular kernels. SIAM J Numer Anal, 41: 364–381 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  14. Vulanovic R. Non-equidistant generalizations of the Gushchin-Shchennikov scheme. Z Angew Math Mec, 67: 625–632 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  15. Li R, Chen Z, Wu W. Generalized Difference Methods for Differential Equations: Numerical Analysis of Finite Volume Methods. New York-Basel: Marcel Dekker Inc, 2000

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  17. O’Malley R E. Introduction to Singular Perturbations. New York: Academic Press, 1974

    MATH  Google Scholar 

  18. Bakhvalov A S. K optimizacii methdov resenia kraevyh zadac prinalicii pogranicnogo sloja. Zh Vychisl Mat Mat Fiz, 9: 841–859 (1969)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Scientific Computing and Computer Applications, Sun Yat-Sen University, Guangzhou, 510275, China

    ZhongYing Chen, ChongNan He & Bin Wu

  2. College of Mathematics and Computer Science, Guangxi University for Nationalities, Nanning, 530006, China

    ChongNan He

Authors
  1. ZhongYing Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. ChongNan He
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Bin Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to ZhongYing Chen.

Additional information

This work was supported by the National Natural Science Foundation of China (Grant Nos. 10771224, 10601070), and the Guangdong Provincial Natural Science Foundation of China (Grant No. 05003308), the MOE Project of Key Research Institute of Humanities and Social Sciences at Universities, and China-France-Russia Mathematics Collaboration (Grant No. 34000-3275100).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Chen, Z., He, C. & Wu, B. High order finite volume methods for singular perturbation problems. Sci. China Ser. A-Math. 51, 1391–1400 (2008). https://doi.org/10.1007/s11425-008-0120-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-008-0120-1

Keywords

MSC(2000)

Search

Navigation