The authors consider a two point boundary value problem of a scalar linear ordinary differential equation (ODE) of second order. In this equation, a parameter implies a singular perturbation problem for small values of the parameter. A Shishkin mesh is used to discretise the domain of dependence following a strategy introduced G. I. Shishkin [Zh. Vychisl. Mat. Fiz. 28, No. 11, 1649–1662 (1988; Zbl 0662.65086)]. Thereby, piecewise equidistant grids are applied, where smaller step sizes arise in the boundary layers of the exact ODE solution.
A finite difference scheme tailored to the ODE is constructed on an arbitrary fitted mesh to obtain a linear system for the numerical approximations. The authors prove that the finite difference method using the Shishkin mesh is uniformly convergent for all . Thereby, the convergence rate is achieved, where denotes the total number of subintervals. In contrast, straightforward techniques exhibit just a rate of . Numerical simulations of five examples, where the exact solution is known, verify the predicted convergence properties.