{
  "abstract": "In this paper, we analyze the standard and the mass-lumped linear finite element methods for the Gross--Pitaevskii eigenvalue problem. For the standard method, we give a new proof of the uniqueness of the discrete ground state up to sign for sufficiently small mesh sizes, which only requires the $H^1$-convergence of the discrete ground states. For both methods, we establish optimal convergence rates in the $L^p$- and $W^{1,p}$-norms for $1 \\le p < \\infty$. Under additional regularity assumptions on the potential, we further obtain an optimal convergence rate in the $W^{1,\\infty}$-norm and an almost optimal convergence rate, up to a logarithmic factor, in the $L^\\infty$-norm. Moreover, we prove that the discrete ground states superconverge to the Ritz projection of the exact ground state in the $W^{1,p}$-seminorm. The analysis is based on a perturbation argument. Numerical experiments confirm the theoretical results.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.41715",
  "author_total": 2,
  "authors": [
    {
      "name": "Tianyang Chu"
    },
    {
      "name": "Mingyan Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/20xxqp3drnx0r.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.41715",
  "fulltext_url": null,
  "id": "20xxqp3drnx0r",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.41715",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2610.41715",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/20xxqp3drnx0r.ris",
  "sources": [
    {
      "first_seen": "2026-11-02T12:21:20Z",
      "native_id": "2610.41715",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.41715"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Numerical analysis of linear finite element methods for the Gross--Pitaevskii eigenvalue problem",
  "title_text": "Numerical analysis of linear finite element methods for the Gross--Pitaevskii eigenvalue problem",
  "type": "preprint",
  "updated_at": "2026-11-06T18:21:01Z",
  "url": "https://research.draconitic.com/paper/20xxqp3drnx0r",
  "venue": "arXiv",
  "versions": []
}