{
  "abstract": "We give an elementary, coordinate-free proof that uniformly inf-sup stable nested Petrov-Galerkin methods on Hilbert spaces satisfy relaxed general quasi-orthogonality. More precisely, for any window of $N$ consecutive levels, the sum of squared Galerkin increments is bounded by the squared error at the beginning of the window times $N^σ$, where $σ<1$. The exponent $σ$ and the prefactor can be chosen explicitly in terms of the uniform projection bound. The proof uses Hilbert-space projection estimates, dyadic iteration, and duality, without relying on matrix factorizations. As a corollary, we obtain explicit growth bounds for finite block-LU factors. In the symmetric indefinite case, the positive and negative spectral splittings of the Galerkin detail spaces yield a characterization of full quasi-orthogonality in terms of uniform boundedness of the associated finite-window sign operators. We also construct a self-adjoint involution and a nested, uniformly inf-sup stable Galerkin sequence for which full general quasi-orthogonality fails. For every $0<α<1$, the construction yields a hierarchy whose finite-window constants have growth of exact order $N^α$, along with an explicit target whose Galerkin increments are not square summable.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.16265",
  "author_total": 1,
  "authors": [
    {
      "name": "Tsogtgerel Gantumur"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8tq96yway0xjb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.16265",
  "fulltext_url": null,
  "id": "8tq96yway0xjb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.16265",
  "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.FA"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "openalex",
      "2206|Advanced Numerical Methods in Computational Mathematics"
    ]
  ],
  "openalex": "W7213408533",
  "pdf_url": "https://arxiv.org/pdf/2609.16265",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8tq96yway0xjb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:35Z",
      "native_id": "2609.16265",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.16265"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Projection geometry and relaxed quasi-orthogonality for inf-sup stable Galerkin methods",
  "title_text": "Projection geometry and relaxed quasi-orthogonality for inf-sup stable Galerkin methods",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:35Z",
  "url": "https://research.draconitic.com/paper/8tq96yway0xjb",
  "venue": "arXiv",
  "versions": []
}