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.
Cite
@article{arxiv.2409.14426,
title = {p and hp Spectral Element Methods for Elliptic Boundary Layer Problems},
author = {Akhlaq Husain and Aliya Kazmi and Subhashree Mohapatra and Ziya Uddin},
journal= {arXiv preprint arXiv:2409.14426},
year = {2025}
}