p and hp Spectral Element Methods for Elliptic Boundary Layer Problems

Abstract: In this article, we propose p and hp least-squares spectral element methods for one-dimensional elliptic boundary layer problems. Stability estimates are derived and we design numerical schemes based on minimizing the residuals in the sense of least-squares in appropriate Sobolev norms. We prove parameter robust uniform error estimates i.e. error in the approximation is independent of the boundary layer parameter. For the p-version we prove a robust uniform convergence rate of O(sqrt(log W)/W), where W denotes the polynomial order used in approximation and for the hp-version the convergence rate is shown to be O(e^-W/logW). Numerical results are presented for a number of model elliptic boundary layer problems confirming the theoretical estimates and uniform convergence results for the p and hp versions.