Numerical analysis of a singularly perturbed nonlinear reaction-diffusion problem with multiple solutions

A nonlinear reaction-diffusion two-point boundary value problem with multiple solutions is considered. Its second-order derivative is multiplied by a small positive parameter ε, which induces boundary layers. Using dynamical systems techniques, asymptotic properties of its discrete sub- and super-solutions are derived. These properties are used to investigate the accuracy of solutions of a standard three-point difference scheme on layer-adapted meshes of Bakhvalov and Shishkin types. It is shown that one gets second-order convergence (with, in the case of the Shishkin mesh, a logarithmic factor) in the discrete maximum norm, uniformly in ε for ε ≤ CN^-1, where N is the number of mesh intervals. Numerical experiments are performed to support the theoretical results.