Basis of splines associated with singularly perturbed advection -diffusion problems

Abstract

Among fitted-operator methods for solving one-dimensional singular perturbation problems one of the most accurate is the collocation by linear combinations of ${1,x,exp{(pm p x)} }$, known as tension spline collocation. There exist well established results for determining the `tension parameter\u27 $p$, as well as special collocation points, that provide higher order local and global convergence rates. However, if the advection-diffusion reaction problem is specified in such a way that two boundary internal layers exist, the method is incapable of capturing only one boundary layer, which happens when no reaction term is present. For a pure advection-diffusion problem we therefore modify the basis accordingly, including only one exponential, i.e. project the solution to the space locally spanned by ${1,x,x^2,exp{(p x)}}$ where $p>0$ is the tension parameter. The aim of the paper is to show that in this situation it is still possible to construct a basis of $C^1$-locally supported functions by a simple knot insertion technique, commonly used in computer aided geometric design. We end by showing that special collocation points can be found, which yield better local and global convergence rates, similar to the tension spline case