A uniformly accurate spline collocation method for a normalized flux

We are concerned with a two-point boundary value problem for a semilinear singularly perturbed reaction–diffusion equation with a singular perturbation parameter ε. Our goal is to construct global ε-uniform approximations of the solution y(x) and the normalized flux P(x)=ε(d/dx)y(x), using the collocation with the classical quadratic splines u(x)∈C1(I) on a slightly modified piecewise uniform mesh of Shishkin type. The constructed approximate solution and normalized flux converge ε-uniformly with the rate O(n−2 ln2 n) and O(n−1 ln n), respectively, on the Shishkin-type mesh, and with O(n−1 ln−2 n) and O(ln−3 n) when the mesh has to be modified. We present numerical experiments in support of these results.