{
  "abstract": "We design and analyse a family of hypocoercivity-preserving fully discrete Galerkin methods for the (inhomogeneous) kinetic Fokker--Planck (kFP) equations, a class of evolution partial differential equations (PDE) with \\emph{degenerate} diffusion. The key idea in the method design is to mimic the framework of enhanced quadratic forms of Villani \\cite{villani} and, thus, to arrive at a coercive bilinear form in a sufficiently strong exponentially weighted norm, admitting spectral gap/Poincar\\'e inequality, despite the degeneracy. The problem is posed as a fourth order evolution PDE on the whole space $\\mathbb{R}^{d}\\times\\mathbb{R}^d$. The spatial discretization employs continuous piecewise polynomial approximation spaces, subordinate to simplicial and/or box-type meshes comprising finite, as well as `infinite' elements. The non-conformity is treated by constructing appropriate numerical fluxes in the spirit of $C^0$-interior penalty ($C^0$-IP) methods for fourth order elliptic problems. The proof of coercivity of the method requires new trace inverse inequalities for exponentially weighted norms on $d$-dimensional simplicial and box-type domains, and also for prismatic domains with infinite length. We prove weighted trace inverse inequalities in all cases and for a variety of exponential weights. Once coercivity is established for the Galerkin method, the proof of exponential convergence to the equilibrium state follows by employing an exponentially weighted Poincar\\'e inequality. A fully discrete extension of the method is also considered by coupling the non-standard spatial discretization with an $hp$-version discontinuous Galerkin time-stepping scheme. The exponential convergence to the equilibrium state for the fully discrete scheme is also proven. The proposed methods preserve the total mass and exhibit \\emph{provably} exponential convergence to the equilibrium state, a manifestation of `numerical hypocoercivity,' making them well suited for long-time kFP simulations. A series of numerical experiments is presented to validate the theoretical results and investigate the convergence behaviour of the proposed method.",
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  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.29621",
  "author_total": 2,
  "authors": [
    {
      "name": "Zhaonan Dong"
    },
    {
      "name": "Emmanuil H Georgoulis"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/apkymthb1q2fj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.29621",
  "fulltext_url": null,
  "id": "apkymthb1q2fj",
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  "landing_url": "https://inria.hal.science/hal-05672281v2",
  "language": "en",
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
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  "metadata_licence": "CC0-1.0",
  "native_subjects": [
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      "2206|Advanced Numerical Methods in Computational Mathematics"
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  "openalex": "W7166696710",
  "pdf_url": "https://inria.hal.science/hal-05672281/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-06-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/apkymthb1q2fj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:03Z",
      "native_id": "hal-05672281",
      "source": "hal",
      "url": "https://inria.hal.science/hal-05672281v2"
    },
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2606.29621",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.29621"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Hypocoercivity-preserving space-time Galerkin methods for kinetic Fokker-Planck equations",
  "title_text": "Hypocoercivity-preserving space-time Galerkin methods for kinetic Fokker-Planck equations",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/apkymthb1q2fj",
  "venue": "HAL",
  "versions": []
}