p and hp Spectral Element Methods for Elliptic Boundary Layer Problems
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866908499596804096 |
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| author | Husain, Akhlaq Kazmi, Aliya Mohapatra, Subhashree Uddin, Ziya |
| author_facet | Husain, Akhlaq Kazmi, Aliya Mohapatra, Subhashree Uddin, Ziya |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_14426 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | p and hp Spectral Element Methods for Elliptic Boundary Layer Problems Husain, Akhlaq Kazmi, Aliya Mohapatra, Subhashree Uddin, Ziya Numerical Analysis Analysis of PDEs 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. |
| title | p and hp Spectral Element Methods for Elliptic Boundary Layer Problems |
| topic | Numerical Analysis Analysis of PDEs |
| url | https://arxiv.org/abs/2409.14426 |