A uniformly convergent method for a singularly perturbed semilinear reaction--diffusion problem with multiple solutions

This paper considers a simple central difference scheme for a singularly perturbed semilinear reaction--diffusion problem, which may have multiple solutions. Asymptotic properties of solutions to this problem are discussed and analyzed. To compute accurate approximations to these solutions, we consider a piecewise equidistant mesh of Shishkin type, which contains $O(N)$ points. On such a mesh, we prove existence of a solution to the discretization and show that it is accurate of order $N^{-2}\ln ^2 N$, in the discrete maximum norm, where the constant factor in this error estimate is independent of the perturbation parameter $\varepsilon$ and $N$. Numerical results are presented that verify this rate of convergence.