The p and hp versions of the nite element method forproblems with boundary layers

We study the uniform approximation of boundary layer functions exp(?x=d) for x 2 (0; 1), d 2 (0; 1], by the p and hp versions of the nite element method. For the p version (with xed mesh), we prove super-exponential convergence in the range p +1=2 > e=(2d). We also establish, for this version, an overall convergence rate of O(p ?1 p lnp) in the energy norm error which is uniform in d, and show that this rate is sharp (up to the p ln p term) when robust estimates uniform in d 2 (0; 1] are considered. For the p version with variable mesh (i.e. the hp version), we show exponential convergence, uniform in d 2 (0; 1], is achieved by taking the rst element at the boundary layer to be of size O(pd). Numerical experiments for a model elliptic singular perturbation problem show good agreement with our convergence estimates even when few degrees of freedom are used and when d is as small as e.g. 10 ?8. They also illustrate the superiority of the hp approach over other methods, including a low order h version with optimal \exponential" mesh reenement. The estimates established in this paper are also applicable in the context of corresponding spectral element methods.