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A uniformly convergent method for a singularly perturbed semilinear reaction-diffusion problem with multiple solutions. (English) Zbl 0858.65081

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.

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
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