Open Access

Monotone finite difference domain decomposition algorithms and applications to nonlinear singularly perturbed reaction-diffusion problems

Advances in Difference Equations20062006:070325

DOI: 10.1155/ADE/2006/70325

Received: 16 September 2004

Accepted: 11 January 2005

Published: 6 February 2006


This paper deals with monotone finite difference iterative algorithms for solving nonlinear singularly perturbed reaction-diffusion problems of elliptic and parabolic types. Monotone domain decomposition algorithms based on a Schwarz alternating method and on box-domain decomposition are constructed. These monotone algorithms solve only linear discrete systems at each iterative step and converge monotonically to the exact solution of the nonlinear discrete problems. The rate of convergence of the monotone domain decomposition algorithms are estimated. Numerical experiments are presented.


Authors’ Affiliations

Institute of Fundamental Sciences, Massey University


  1. Boglaev I: A numerical method for a quasilinear singular perturbation problem of elliptic type. USSR Computational Mathematics and Mathematical Physics 1988, 28: 492–502.MathSciNetMATHGoogle Scholar
  2. Boglaev I: Numerical solution of a quasilinear parabolic equation with a boundary layer. USSR Computational Mathematics and Mathematical Physics 1990,30(3):55–63. 10.1016/0041-5553(90)90190-4MathSciNetView ArticleMATHGoogle Scholar
  3. Boglaev I: Monotone iterative algorithms for a nonlinear singularly perturbed parabolic problem. Journal of Computational and Applied Mathematics 2004,172(2):313–335. 10.1016/ ArticleMATHGoogle Scholar
  4. Boglaev I: On monotone iterative methods for a nonlinear singularly perturbed reaction-diffusion problem. Journal of Computational and Applied Mathematics 2004,162(2):445–466. 10.1016/ ArticleMATHGoogle Scholar
  5. Ladyženskaja OA, Solonnikov VA, Ural'ceva NN: Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs. Volume 23. Izdat. "Nauka", Moscow; 1968:xi+648.Google Scholar
  6. Ladyženskaja OA, Ural'ceva NN: Linear and Quasi-Linear Elliptic Equations. Academic Press, New York; 1968.Google Scholar
  7. Miller JJH, O'Riordan E, Shishkin GI: Fitted Numerical Methods for Singular Perturbation Problems. World Scientific, New Jersey; 1996:xiv+166.View ArticleMATHGoogle Scholar
  8. Pao CV: Monotone iterative methods for finite difference system of reaction-diffusion equations. Numerische Mathematik 1985,46(4):571–586. 10.1007/BF01389659MathSciNetView ArticleMATHGoogle Scholar
  9. Pao CV: Finite difference reaction diffusion equations with nonlinear boundary conditions. Numerical Methods for Partial Differential Equations. An International Journal 1995,11(4):355–374. 10.1002/num.1690110405MathSciNetView ArticleMATHGoogle Scholar
  10. Saad Y, Schultz MH: GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM Journal on Scientific and Statistical Computing 1986,7(3):856–869. 10.1137/0907058MathSciNetView ArticleMATHGoogle Scholar
  11. Samarskii AA: The Theory of Difference Schemes, Monographs and Textbooks in Pure and Applied Mathematics. Volume 240. Marcel Dekker, New York; 2001:xviii+761.Google Scholar


© Boglaev and Hardy 2006

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.