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arXiv · 2409.14426

p and hp Spectral Element Methods for Elliptic Boundary Layer Problems

Akhlaq Husain·Aliya Kazmi·Subhashree Mohapatra·Ziya Uddin
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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.

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Akhlaq Husain, Aliya Kazmi, Subhashree Mohapatra, Ziya Uddin. 2024-09-22. p and hp Spectral Element Methods for Elliptic Boundary Layer Problems. https://arxiv.org/abs/2409.14426

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