{
  "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 $O\\left(\\frac{\\log W}{W^2}\\right)$ for the $p$-version and at the rate $O(εα^{2W})$, uniformly in $ε$, for the $rp$-version, where $0<ε\\leq1$ 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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.29185",
  "author_total": 3,
  "authors": [
    {
      "name": "Sonia"
    },
    {
      "name": "Akhlaq Husain"
    },
    {
      "name": "Arshad Khan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5abdtj0z9hbfj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.29185",
  "fulltext_url": null,
  "id": "5abdtj0z9hbfj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.29185",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "openalex",
      "2206|Advanced Numerical Methods in Computational Mathematics"
    ]
  ],
  "openalex": "W7214293551",
  "pdf_url": "https://arxiv.org/pdf/2609.29185",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5abdtj0z9hbfj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.29185",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.29185"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "$p$ and $rp$-Spectral Element Methods for Two-dimensional Elliptic Boundary Layer Problems",
  "title_text": "p and rp-Spectral Element Methods for Two-dimensional Elliptic Boundary Layer Problems",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/5abdtj0z9hbfj",
  "venue": "arXiv",
  "versions": []
}