{
  "abstract": "Low-order refined (LOR) preconditioning replaces a high-order operator with a low-order discretisation on a refined nodal mesh. For tensor-product elements, the two operators are spectrally equivalent with bounds independent of the polynomial order $P$, but the construction does not extend directly to simplex and mixed-element discretisations. This work makes two contributions: it extends LOR preconditioning to simplex and mixed-element discretisations, including triangular, tetrahedral, and prismatic elements, and establishes a generalised Vandermonde transformation linking the modal and nodal LOR formulations, showing that the resulting preconditioned spectra and Krylov convergence are independent of the high-order basis. Numerical experiments show controlled iteration growth on triangular meshes despite increasing condition number, and controlled iteration counts up to $P=5$ on tetrahedral, prismatic, and mixed-element meshes. A single algebraic multigrid V-cycle per outer iteration gives the best balance of iteration count and cost. The method is applied to a production incompressible Navier-Stokes simulation of a race-car front-wing and wheel configuration, discretised on a mesh of $2.87\\times10^6$ mixed prismatic and tetrahedral elements giving $32.2\\times10^6$ pressure degrees of freedom at $P=3$. LOR reduces the mean pressure conjugate gradient (CG) iteration count from $235.3$ to $5.5$, and the pressure-solve time over 1000 timesteps by 16.1%, relative to the default production static-condensation diagonal preconditioner in Nektar++. The one-time cost of constructing the LOR preconditioner is amortised over the production simulation.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.31443",
  "author_total": 6,
  "authors": [
    {
      "name": "Parv Khurana"
    },
    {
      "name": "Henrik Wüstenberg"
    },
    {
      "name": "David Moxey"
    },
    {
      "name": "Athanasios Chatzopoulos"
    },
    {
      "name": "Julien Hoessler"
    },
    {
      "name": "Spencer J. Sherwin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8c6bwdtmb8qpg.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.31443",
  "fulltext_url": null,
  "id": "8c6bwdtmb8qpg",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.31443",
  "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": "W7214655212",
  "pdf_url": "https://arxiv.org/pdf/2609.31443",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8c6bwdtmb8qpg.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:07Z",
      "native_id": "2609.31443",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.31443"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Low-Order Refined Preconditioning for Spectral/hp Element Method for Complex, 3D Geometries",
  "title_text": "Low-Order Refined Preconditioning for Spectral/hp Element Method for Complex, 3D Geometries",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:07Z",
  "url": "https://research.draconitic.com/paper/8c6bwdtmb8qpg",
  "venue": "arXiv",
  "versions": []
}