{
  "abstract": "We introduce and analyze a barrier-regularized symmetric Nitsche method for the scalar Signorini problem. Applying a logarithmic barrier to the nonnegative slack variable in an augmented Lagrangian and then eliminating that variable yields a smooth positive-part operator and the perturbed complementarity relation on a primal-dual central path, with barrier parameter $μ=γs$. For every $s>0$, the discrete problem is continuously differentiable, uniquely solvable, and has a symmetric positive definite Newton matrix when the Nitsche parameter is sufficiently large. Under a mild barrier-feasibility condition, the continuous logarithmic energy has a unique minimizer $u_μ$ for every $μ>0$, its gap is positive almost everywhere, and $\\|u-u_μ\\|_{H^1(Ω)}\\lesssimμ^{1/2}$. Under the Sobolev regularity used for finite element approximation, this minimizer satisfies the \\(L^2\\) central-path boundary law. If the obstacle is locally the trace of an $H^2$-function, we also prove a uniform local $H^2$ bound in the interior of each planar contact face, on neighborhoods that may contain a free-boundary point of the limiting Signorini solution. The method is exactly consistent and quasi-optimal relative to $u_μ$, with constants independent of $s$. If $u_μ$ is uniformly bounded in $H^r(Ω)$, $3/2<r\\leq k+1$, then $\\|u-u_h\\|_{H^1(Ω)}\\lesssim h^{r-1}\\|u_μ\\|_{H^r(Ω)}+μ^{1/2}$; hence the sufficient balance $s\\lesssim h^{2r-1}$ preserves the available energy-norm rate. We also derive rates for discrete penetration and the complementarity residual. Numerical experiments with $P_1$ and $P_2$ elements on structured and unstructured meshes support the predicted rates and the local Newton theory, while showing that a rate-preserving smoothing may still resolve the contact set poorly.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.30470",
  "author_total": 2,
  "authors": [
    {
      "name": "Peter Hansbo"
    },
    {
      "name": "Mats G. Larson"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2pkqmeds13pdb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.30470",
  "fulltext_url": null,
  "id": "2pkqmeds13pdb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.30470",
  "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",
      "1703|Contact Mechanics and Variational Inequalities"
    ]
  ],
  "openalex": "W7204955079",
  "pdf_url": "https://arxiv.org/pdf/2608.30470",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-31",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2pkqmeds13pdb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2608.30470",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.30470"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "A Barrier-Regularized Symmetric Nitsche Method for the Signorini Problem",
  "title_text": "A Barrier-Regularized Symmetric Nitsche Method for the Signorini Problem",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/2pkqmeds13pdb",
  "venue": "arXiv",
  "versions": []
}