Available online at www.sciencedirect.com Mathematics and Computers in Simulation 79 (2009) 2490–2505 The discrete minimum principle for quadratic spline discretization of a singularly perturbed problem K. Surla a , Z. Uzelac b,∗ , Lj. Teofanov b a Faculty of Sciences, University of Novi Sad, Trg D. Obradovi´ ca 4, 21000 Novi Sad, Serbia b Faculty of Technical Sciences, University of Novi Sad, Trg D. Obradovi´ ca 6, 21000 Novi Sad, Serbia Available online 31 January 2009 Abstract We consider a singularly perturbed boundary value problem with two small parameters. The problem is numerically treated by a quadratic spline collocation method. The suitable choice of collocation points provides the discrete minimum principle. Error bounds for the numerical approximations are established. Numerical results give justification of the parameter-uniform convergence of the numerical approximations. © 2009 IMACS. Published by Elsevier B.V. All rights reserved. Keywords: Singular perturbation; Convection–diffusion problems; Two small parameters; Shishkin mesh; Spline difference schemes 1. Introduction We consider the two-parameter singularly perturbed convection–diffusion–reaction boundary value problem Ly := εy ′′ (x) + μa(x)y ′ (x) - b(x)y(x) = f (x), x ∈ (0, 1), y(0) = γ 0 , y(1) = γ 1 , (1) with two small parameters 0 < ε, μ ≪ 1. The functions a,b and f are assumed to be sufficiently smooth and A ≥ a(x) ≥ a> 0, b(x) ≥ β> 0, x ∈ [0, 1]. Under these assumptions the problem (1) has a unique solution which exhibits exponential boundary layers at x = 0 and x = 1 [3,5,7]. Boundary layers are regions where the solution and its derivatives vary rapidly. Most of the traditional numerical methods fail to catch those rapid changes of the solution, and this failure in turn pollutes the numerical approximation on the whole domain. For the construction of any numerical method solving singularly perturbed problem it is crucial to have information about the behavior of derivatives of the exact solution. The bounds on derivatives are required in the mesh refinement strategy as well as in the error analysis. Layer-adapted meshes are usually used to solve singularly perturbed problems. We use a piecewise uniform Shishkin mesh which can be chosen a priori when one has some knowledge of the structure of these layers. When μ = 1, the problem (1) becomes a convection–diffusion problem with the boundary layer of width O(ε) in the neighborhood of x = 0. If μ = 0, we have the reaction–diffusion problem with boundary layers of width O( √ ε) at x = 0 and x = 1, which is treated by spline collocation method in [8]. For the asymptotic structure of the solutions ∗ Corresponding author. E-mail addresses: ksurla@im.ns.ac.yu (K. Surla), zora@uns.ns.ac.yu (Z. Uzelac), ljiljap@uns.ns.ac.yu (Lj. Teofanov). 0378-4754/$36.00 © 2009 IMACS. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.matcom.2009.01.007