Numerical analysis of singularly perturbed nonlinear reaction-diffusion problems 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 suband super-solutions are derived. These properties are used to investigate the accuracy of solutions of a standard three-point difference scheme on layeradapted 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. AMS Subject Classification: Primary 65L10, Secondary 65L12, 65L50.