p and rp-Spectral Element Methods for Two-dimensional Elliptic Boundary Layer Problems
Abstract
We propose p and rp-spectral element methods for elliptic boundary layer problems on two dimensional rectangular domains. To resolve boundary layers, we use p-version with a fixed mesh and an rp-version with a boundary layer mesh consisting of thin needle-like elements near the boundary layer and coarse elements away from the layer. Stability estimates are derived using non-conforming spectral element functions. The numerical scheme for both the methods is based on minimising a least-squares functional in appropriate Sobolev norms. We construct robust preconditioners to manage the condition number of the normal equations derived from the least-squares formulation. The method is able to approximate boundary layers at a rate O((log W)/W2) for the p-version and at the rate O(εα2W), uniformly in ε, for the rp-version, where 0<ε≤1 is the boundary layer parameter, α<1 is a constant and W denotes the degree of the approximating polynomial. Numerical results are provided for model elliptic boundary layer problems for a range of boundary layer parameters. Simulation results demonstrate the efficiency and robustness of the method in capturing boundary layers on rectangular domains.
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BibTeX
@misc{sonia2026rpspectral,
title = {{$p$ and $rp$-Spectral Element Methods for Two-dimensional Elliptic Boundary Layer Problems}},
author = {Sonia and Akhlaq Husain and Arshad Khan},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.29185},
eprint = {2609.29185},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.29185},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - p and rp-Spectral Element Methods for Two-dimensional Elliptic Boundary Layer Problems
AU - Sonia
AU - Akhlaq Husain
AU - Arshad Khan
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.29185
UR - https://arxiv.org/abs/2609.29185
AB - We propose p and rp-spectral element methods for elliptic boundary layer problems on two dimensional rectangular domains. To resolve boundary layers, we use p-version with a fixed mesh and an rp-version with a boundary layer mesh consisting of thin needle-like elements near the boundary layer and coarse elements away from the layer. Stability estimates are derived using non-conforming spectral element functions. The numerical scheme for both the methods is based on minimising a least-squares functional in appropriate Sobolev norms. We construct robust preconditioners to manage the condition number of the normal equations derived from the least-squares formulation. The method is able to approximate boundary layers at a rate O((log W)/W^2) for the p-version and at the rate O(εα^2W), uniformly in ε, for the rp-version, where 0<ε≤1 is the boundary layer parameter, α<1 is a constant and W denotes the degree of the approximating polynomial. Numerical results are provided for model elliptic boundary layer problems for a range of boundary layer parameters. Simulation results demonstrate the efficiency and robustness of the method in capturing boundary layers on rectangular domains.
ER -