Uniformly convergent schemes for singularly perturbed differential equations based on collocation methods

It is well known that a polynomial-based approximation scheme applied to a singularly perturbed equation is not uniformly convergent over the geometric domain of study. Such scheme results in a numerical solution, say σ which suffers from severe inaccuracies particularly in the boundary layer. What we say in the current paper is this: when one uses a grid which is not too coarse the resulted solution, even being nonuniformly convergent may be used in an iterated scheme to get a good approximation solution that is uniformly convergent over the whole geometric domain of study. In this paper, we use the collocation method as model of polynomial approximation. We start from a precise localization of the boundary layer then we decompose the domain of study, say Ω into the boundary layer, say Ω ϵ and its complementary Ω 0 . Next we go to the heart of our work which is to make a repeated use of the collocation method. We show that the second generation of polynomial approximation is convergent and it yields an improved error bound compared to those usually appearing in the literature.