Skip to main content
SpringerLink
Log in

Optimal L ∞ estimates for Galerkin methods for nonlinear singular two-point boundary value problems

  • Published:
Acta Mathematicae Applicatae Sinica, English Series Aims and scope Submit manuscript

Abstract

In this paper, we apply the symmetric Galerkin methods to the numerical solutions of a kind of singular linear two-point boundary value problems. We estimate the error in the maximum norm. For the sake of obtaining full superconvergence uniformly at all nodal points, we introduce local mesh refinements. Then we extend these results to a class of nonlinear problems. Finally, we present some numerical results which confirm our theoretical conclusions.

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. Chen, C.M. Superconvergence of Galerkin solutions for singular nonlinear two-point boundary value problems. Math. Numer. Sinica., 7(2): 113–123 (1985) (in Chinese)

    MathSciNet  Google Scholar 

  2. Ciarlet, P.G., Natterer, F., Varga, R.S. Numerical methods of high-order accuracy for singular nonlinear boundary value problems. Numer. Math., 15: 87–99 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  3. Ciarlet, P.G., Schultz, M.H., Varga, R.S. Numerical methods of high-order accuracy for nonlinear boundary value problems. I. One dimensional problem. Numer. Math., 9: 394–430 (1966/1967)

    Article  MathSciNet  MATH  Google Scholar 

  4. Eisenstat, S.G., Schreiber, R.S., Schultz, M.H. Finite element methods for spherically symmetric elliptic equations. Yale Computer Science Report, 109 (1977)

  5. Eriksson, K., Nie, Y.Y. Convergence analysis for a nonsymmetric Galerkin method for a class of singular boundary value problems in one space dimension. Math. Comp., 49(179): 167–186 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  6. Eriksson, K., Thomée, V. Galerkin method for singular boundary value problems in one space dimension. Math. Comput., 42: 345–367 (1984)

    Article  MATH  Google Scholar 

  7. Jespersen, D. Ritz-Galerkin methods for singular boundary value problems. SIAM J. Numer. Anal., 15: 813–834 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  8. Noor, M.A., Whiteman, J.R. Error bounds for finite element solutions of mildly nonlinear elliptic boundary value problems. Numer. Math., 26(1): 107–116 (1976)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, China

    Xu Zhang

  2. LSEC, ICMSEC, Academy of Mathematics and Systems Science, CAS, Beijing, 100190, China

    Zhong-ci Shi

Authors
  1. Xu Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhong-ci Shi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xu Zhang.

Additional information

Supported by the Scientific Research Foundation for the Doctor, Nanjing University of Aeronautics and Astronautics (No. 1008-907359).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, X., Shi, Zc. Optimal L ∞ estimates for Galerkin methods for nonlinear singular two-point boundary value problems. Acta Math. Appl. Sin. Engl. Ser. 31, 719–728 (2015). https://doi.org/10.1007/s10255-015-0498-9

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10255-015-0498-9

Keywords

2000 MR Subject Classification

Search

Navigation