A uniformly accurate spline collocation method for a normalized flux

Abstract

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/d x) y( x), using the collocation with the classical quadratic splines u( x)∈ C 1( 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 ln 2 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.