Skip to main content
Springer Nature Link
Log in

Monotone Iterates for Solving Nonlinear Monotone Difference Schemes

  • Published:
Computing Aims and scope Submit manuscript

Abstract

This paper is concerned with monotone iterative algorithms for solving nonlinear monotone difference schemes of elliptic type. Firstly, the monotone method (known as the method of lower and upper solutions) is applied to computing the nonlinear monotone difference schemes in the canonical form. Secondly, a monotone domain decomposition algorithm based on a modification of the Schwarz alternating method is constructed. This monotone algorithm solves only linear discrete systems at each iterative step and converges monotonically to the exact solution of the nonlinear monotone difference schemes. Numerical experiments are presented.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

References

  • A. Samarskii (2001) The theory of difference schemes Marcel Dekker New York Basel Occurrence Handle0971.65076

    MATH  Google Scholar 

  • J. M. Ortega W. C. Rheinboldt (1970) Iterative solution of nonlinear equations in several variables Academic Press New York London Occurrence Handle0949.65053

    MATH  Google Scholar 

  • C. V. Pao (1985) ArticleTitleMonotone iterative methods for finite difference system of reaction-diffusion equations Numer. Math. 46 571–586 Occurrence Handle0589.65072 Occurrence Handle796645 Occurrence Handle10.1007/BF01389659

    Article  MATH  MathSciNet  Google Scholar 

  • C. V. Pao (2003) ArticleTitleAccelerated monotone iterations for numerical solutions of nonlinear elliptic boundary value problems Comput. Math. Appl. 46 1535–1544 Occurrence Handle1057.65025 Occurrence Handle2024227 Occurrence Handle10.1016/S0898-1221(03)90189-1

    Article  MATH  MathSciNet  Google Scholar 

  • R. Nabben (2003) ArticleTitleComparisons between multiplicative and additive Schwarz iterations in domain decomposition methods Numer. Math. 95 145–162 Occurrence Handle1026.65021 Occurrence Handle1993942 Occurrence Handle10.1007/s00211-002-0444-7

    Article  MATH  MathSciNet  Google Scholar 

  • R. S. Varga (2000) Matrix iterative analysis Springer Berlin Heidelberg Occurrence Handle0998.65505

    MATH  Google Scholar 

  • I. Boglaev (2004) ArticleTitleOn monotone iterative methods for a nonlinear singularly perturbed reaction-diffusion problem J. Comput. Appl. Math. 162 445–466 Occurrence Handle1041.65078 Occurrence Handle2028040 Occurrence Handle10.1016/j.cam.2003.08.035

    Article  MATH  MathSciNet  Google Scholar 

  • E. Bohl (1981) Finite Modelle gewöhnlicher Randwertaufgaben Teubner Stuttgart Occurrence Handle0472.65070

    MATH  Google Scholar 

  • I. Boglaev (1988) ArticleTitleA numerical method for a quasi–linear singular perturbation problem of elliptic type USSR Comput. Maths. Math. Phys. 28 492–502 Occurrence Handle0671.65079 Occurrence Handle943610

    MATH  MathSciNet  Google Scholar 

  • J. J. H. Miller E. O'Riordan G. I. Shishkin (1996) Fitted numerical methods for singular perturbation problems World Scientific Singapore Occurrence Handle0915.65097

    MATH  Google Scholar 

  • Y. Saad M. H. Schultz (1986) ArticleTitleGMRES: A generalized minimal residual method for solving nonsymmetric linear systems SIAM J. Sci. Stat. Comput. 7 856–869 Occurrence Handle0599.65018 Occurrence Handle848568 Occurrence Handle10.1137/0907058

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Fundamental Sciences, Massey University, Palmerston North, New Zealand

    I. Boglaev

Authors
  1. I. Boglaev
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to I. Boglaev.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Boglaev, I. Monotone Iterates for Solving Nonlinear Monotone Difference Schemes. Computing 78, 17–30 (2006). https://doi.org/10.1007/s00607-006-0168-0

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00607-006-0168-0

AMS Subject Classifications

Keywords