{
  "abstract": "For $1<p<2$, the $p$-Laplace equation and its $p$-Stokes generalization are difficult to solve numerically. Newton's method converges rapidly only close to the solution, with iteration counts that grow under mesh refinement and deteriorate as $p\\to1$. The more robust Picard iteration converges only linearly. Rather than globalizing or preconditioning Newton's method, we modify the system to which it is applied. We \\emph{lift} the equation by introducing the flux $|\\nabla u|^{p-2}\\nabla u$ as an auxiliary (or ''lifting'') variable, apply a nonlinear \\emph{transformation} to the resulting constitutive relation, \\emph{linearize}, and \\emph{eliminate} the auxiliary variable by static condensation. Lifting alone leaves the linearization unchanged; it is the preceding transformation that yields the new method. The elimination is algebraic and pointwise at the quadrature points, so the flux variable is never discretized, no inf-sup condition or indefinite system arises, and the cost per iteration is that of a standard Newton step. Together with a pointwise feasibility bound on the lifting variable that keeps the diffusion tensor uniformly positive definite, this yields an iteration that we prove, in finite dimensions and for the $p$-Laplace equation, to converge globally and locally at a quadratic rate; a one-dimensional model problem explains why the lagged flux variable removes the zig-zag behavior of standard Newton for $p$ close to one. Firedrake-based experiments for $p$-Laplace problems in two and three dimensions and for stationary and time-dependent $p$-Stokes flows show iteration counts largely insensitive to $p$ and to mesh refinement, and up to an order of magnitude fewer iterations than standard Newton for $p$ close to one.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.12241",
  "author_total": 4,
  "authors": [
    {
      "name": "Gonzalo G. De Diego"
    },
    {
      "name": "Fortino Garcia"
    },
    {
      "name": "Georg Stadler"
    },
    {
      "name": "Kewei Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/09f7t6qyc6e4x.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.12241",
  "fulltext_url": null,
  "id": "09f7t6qyc6e4x",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.12241",
  "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"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "2206|Advanced Numerical Methods in Computational Mathematics"
    ]
  ],
  "openalex": "W7212550540",
  "pdf_url": "https://arxiv.org/pdf/2609.12241",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/09f7t6qyc6e4x.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:11Z",
      "native_id": "2609.12241",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.12241"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Transform before linearizing: robust Newton methods for singular $p$-Laplace and $p$-Stokes equations",
  "title_text": "Transform before linearizing: robust Newton methods for singular p-Laplace and p-Stokes equations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:11Z",
  "url": "https://research.draconitic.com/paper/09f7t6qyc6e4x",
  "venue": "arXiv",
  "versions": []
}