Uniformly convergent schemes for singularly perturbed differential equations based on collocation methods
The author considers initial value problems for singularly perturbed equations of arbitrary order \(m\) of the form: \( \varepsilon u^{(m)} + \sum_{i=0}^{m-1} a_i u^{(i)} = f \) with \( x \in (a,b) \), stating that they are known as diffusion-convection problems. However in the standard literature these kind of problems are usually described by boundary value problems for second order equations and there is a large number of publications on their numerical solution on non uniform grids. Concerning the aim of the paper the author states: ``We start from a precise localization of the boundary layer then we decompose the domain of study, say \( \Omega \) into the boundary layer, say \( \Omega_{\varepsilon} \) and its complementary \(\Omega_0\). 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 with those usually appearing in the literature''.
scientific article; zbMATH DE number 1533310