{
  "abstract": "In this paper, we develop a preconditioner that is robust with respect to both the mesh size and the interaction horizon for finite element discretizations of volume-constrained nonlocal diffusion problems. Motivated by the Fourier multiplier of the nonlocal operator, we construct the preconditioner $B_h=G_h+βδ^2D_h^{-1}$, where $G_h$ is a symmetric local Poisson solver, $D_h$ is the lumped mass matrix, and $β$ is a kernel-dependent constant. We rigorously prove that the spectrum of the $B_h$-preconditioned system is bounded above and away from zero uniformly with respect to the mesh size and horizon. Moreover, we develop a pair-cell assembly for the finite element matrix that preserves the symmetry and positive definiteness required by the preconditioner and PCG, and establish an $L^2$-error estimate for the finite element discretization. Finally, we use a single symmetric geometric multigrid V-cycle for $G_h$ in the numerical experiments, and the corresponding numerical results confirm the predicted convergence rate and show bounded condition numbers and PCG iteration counts over the tested meshes, horizons, and kernels.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.24143",
  "author_total": 4,
  "authors": [
    {
      "name": "Jiashu Lu"
    },
    {
      "name": "Yufeng Nie"
    },
    {
      "name": "Haolun Zhang"
    },
    {
      "name": "Pingrui Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7ad94papsyf1y.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.24143",
  "fulltext_url": null,
  "id": "7ad94papsyf1y",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.24143",
  "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": "W7214021227",
  "pdf_url": "https://arxiv.org/pdf/2609.24143",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7ad94papsyf1y.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.24143",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.24143"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "A mesh- and horizon-robust preconditioner for finite element discretizations of volume-constrained nonlocal diffusion problems",
  "title_text": "A mesh- and horizon-robust preconditioner for finite element discretizations of volume-constrained nonlocal diffusion problems",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/7ad94papsyf1y",
  "venue": "arXiv",
  "versions": []
}