The $p$ and $hp$ versions of the finite element method for problems with boundary layers

We study the uniform approximation of boundary layer functions $\exp (-x/d)$ for $x\in (0,1)$, $d\in (0,1]$, by the $p$ and $hp$ versions of the finite element method. For the $p$ version (with fixed 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 ${\mathcal O}(p^{-1}\sqrt {\ln p})$ in the energy norm error which is uniform in $d$, and show that this rate is sharp (up to the $\sqrt {\ln p}$ term) when robust estimates uniform in $d\in (0,1]$ are considered. For the $p$ version with variable mesh (i.e., the $hp$ version), we show that exponential convergence, uniform in $d\in (0,1]$, is achieved by taking the first element at the boundary layer to be of size ${\mathcal O}(pd)$. 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 refinement. The estimates established in this paper are also applicable in the context of corresponding spectral element methods.