{
  "abstract": "We study minimal residual space-time finite element discretizations of linear parabolic initial value problems in canonical space-time variational form. To deal with the arising dual norm, we introduce the Riesz lift of the residual as an additional variable. Quasi-optimality of the primal variable of the mixed system follows from a uniform inf-sup condition. This condition is known to be satisfied for finite element spaces w.r.t. prismatic partitions of the space-time cylinder that allow for a decomposition into time-slabs. We prove that this condition cannot be expected to hold otherwise. To recover stability for general partitions and the data at hand, we derive an a posteriori condition on the error between the exact Riesz lift of the residual and its Galerkin approximation -- being the secondary variable of our system -- under which the primal variable is quasi-optimal. We derive a posteriori error estimators for both variables, and use them in a double-adaptive loop that alternates test-space with trial-space enrichment. We illustrate our findings with numerical experiments in $1+1$ and $2+1$ dimensions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.01748",
  "author_total": 3,
  "authors": [
    {
      "name": "Gregor Gantner"
    },
    {
      "name": "Robin Smeets"
    },
    {
      "name": "Rob Stevenson"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3m0fsxpr44pzf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.01748",
  "fulltext_url": null,
  "id": "3m0fsxpr44pzf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.01748",
  "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": "W7207587303",
  "pdf_url": "https://arxiv.org/pdf/2609.01748",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3m0fsxpr44pzf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:20Z",
      "native_id": "2609.01748",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.01748"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "A Double-Adaptivity Solver for Parabolic PDEs",
  "title_text": "A Double-Adaptivity Solver for Parabolic PDEs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:20Z",
  "url": "https://research.draconitic.com/paper/3m0fsxpr44pzf",
  "venue": "arXiv",
  "versions": []
}