On the $hp$ Finite Element Method for the One Dimensional Singularly Perturbed Convection-Diffusion Problems

Abstract

In this work, a singularly perturbed two-point boundary value problem of convection-diffusion type is considered. An $hp$ version finite element method on a strongly graded piecewise uniform mesh of Shishkin type is used to solve the model problem. With the analytic assumption of the input data, it is shown that the method converges exponentially and the convergence is uniformly valid with respect to the singular perturbation parameter.