A note on iterative methods for solving singularly perturbed problems using non-monotone methods on Shishkin meshes Ali R. Ansari * , Alan F. Hegarty Department of Mathematics and Statistics College of Informatics, University of Limerick, Limerick, Ireland Received 19 July 2002; received in revised form 9 December 2002; accepted 14 May 2003 Abstract Non-monotone methods with Shishkin meshes are employed in obtaining finite difference schemes for solving a linear two-dimensional steady state convection–diffusion problem. Preconditioners are used that significantly reduce the number of iterations of the linear solver. Computational results for a Galerkin method are presented which indicate parameter robust, super-linear orders of convergence. Ó 2003 Elsevier B.V. All rights reserved. Keywords: Preconditioning; Iterative solvers; Shishkin meshes; Non-monotone methods 1. Introduction We consider the efficient use of Krylov space iterative solvers, such as CGS [2], BiCGStab [3] and GMRES [4] for the solution of the large sparse linear systems which arise in the course of the numerical solution of two-dimensional linear convection–diffusion equations, using Shishkin meshes [1–3]. Efficient implementation of these methods requires the use of appropriate preconditioners, usually involving some incomplete factorisation of the coefficient matrix. We are particularly concerned with the difficulties which arise when non-monotone difference operators are used, in which case standard preconditioners have not proved at all practicable. We will show that minor modification of these preconditioners does indeed yield efficient iterative techniques for these problems. The model problem we wish to solve here is the linear steady-state convection–diffusion problem Lu  eDu þ a ru ¼ f in X ¼ð0; 1Þð0; 1Þ; u ¼ g on oX; a ¼ða 1 ; a 2 Þ; a i P a i > 0; i ¼ 1; 2; ð1Þ * Corresponding author. Fax: +61-334927. E-mail addresses: ali.ansari@ul.ie (A.R. Ansari), alan.hegarty@ul.ie (A.F. Hegarty). 0045-7825/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0045-7825(03)00369-4 Comput. Methods Appl. Mech. Engrg. 192 (2003) 3673–3687 www.elsevier.com/locate/cma