Sun, Guangfu; Stynes, Martin A uniformly convergent method for a singularly perturbed semilinear reaction-diffusion problem with multiple solutions. (English) Zbl 0858.65081 Math. Comput. 65, No. 215, 1085-1109 (1996). The authors considers the semilinear boundary value problem \[ -\varepsilon^2 u''(x) + b(x,u)=0, \quad u(0)= u(1)=0, \] where \(\varepsilon\) is a small positive parameter, under the assumption that it has a stable reduced solution and stable boundary layers. A uniformly convergent method for this problem is described. From the theoretical point of view the techniques of super and sub solutions and degree theory are used. A central difference scheme on a Shishkin mesh, which is piecewise equidistant, is proposed for the numerical solution. This method is uniformly convergent of order \(N^{-2} \ln^2N\) with \(\varepsilon \leq N^{-1}\). Numerical results that confirm the uniform accuracy are presented. Reviewer: Z.Schneider (Bratislava) Cited in 15 Documents MSC: 65L10 Numerical solution of boundary value problems involving ordinary differential equations 65L12 Finite difference and finite volume methods for ordinary differential equations 65L50 Mesh generation, refinement, and adaptive methods for ordinary differential equations 34B15 Nonlinear boundary value problems for ordinary differential equations 34E15 Singular perturbations for ordinary differential equations Keywords:uniform convergence; numerical examples; semilinear boundary value problem; boundary layers; super and sub solutions; central difference scheme; Shishkin mesh PDFBibTeX XMLCite \textit{G. Sun} and \textit{M. Stynes}, Math. Comput. 65, No. 215, 1085--1109 (1996; Zbl 0858.65081) Full Text: DOI