{
  "abstract": "Rigorous error analysis for numerical approximations of parabolic equations with hysteresis remains limited, even for the widely used Prandtl--Ishlinskii model of play type. In this work, we establish an $O(h+τ)$ error bound for an implicit Euler $P_1$ finite element discretization. The analysis requires neither higher-order temporal regularity nor additional spatial regularity of the hysteresis variables. For the temporal discretization, we exploit a convex subgradient-flow structure in a weighted Hilbert space and estimates of residual monotone terms to obtain first-order convergence. For the spatial discretization, only the diffusive field is restricted to the finite element space, and a constraint-preserving comparison yields an $O(h)$ semidiscrete estimate. The analysis is developed for play-type Prandtl--Ishlinskii operators formulated directly on a spatial Hilbert space, encompassing the canonical pointwise extension as well as more general spatially structured constraints.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.23676",
  "author_total": 2,
  "authors": [
    {
      "name": "Shu Xu"
    },
    {
      "name": "Liqun Cao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fpgkpaj436w9j.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.23676",
  "fulltext_url": null,
  "id": "fpgkpaj436w9j",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.23676",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "openalex",
      "2206|Advanced Numerical Methods in Computational Mathematics"
    ]
  ],
  "openalex": "W7214044920",
  "pdf_url": "https://arxiv.org/pdf/2609.23676",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fpgkpaj436w9j.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2609.23676",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.23676"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Numerical analysis of parabolic equations with Prandtl--Ishlinskii hysteresis of play type",
  "title_text": "Numerical analysis of parabolic equations with Prandtl--Ishlinskii hysteresis of play type",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/fpgkpaj436w9j",
  "venue": "arXiv",
  "versions": []
}