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A structure-preserving finite volume scheme for the time-fractional Allen-Cahn equation with nondegenerate state-dependent mobility — arXiv:2611.00837

Source: arXiv (en)
Original: https://arxiv.org/abs/2611.00837
License (as archived): CC0-1.0
Archived: 2026-11-03T18:48:31.044Z
SHA-256 of the archived text: 8fd9500f12245cd20792ded54866b837c57b7e061b0eb497997d807db4c9945a
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How to cite

A structure-preserving finite volume scheme for the time-fractional Allen-Cahn equation with nondegenerate state-dependent mobility — arXiv:2611.00837 — arXiv (en), https://arxiv.org/abs/2611.00837 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_6db2600a56f9 (Metraipolis archive dc_6db2600a56f9, sha256 8fd9500f12245cd20792ded54866b837c57b7e061b0eb497997d807db4c9945a)

BibTeX

@misc{dc6db2600a56f9,
  title = {A structure-preserving finite volume scheme for the time-fractional Allen-Cahn equation with nondegenerate state-dependent mobility — arXiv:2611.00837},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2611.00837},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_6db2600a56f9; sha256 8fd9500f12245cd20792ded54866b837c57b7e061b0eb497997d807db4c9945a},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2611.00837. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

We construct and analyze a fully discrete finite volume scheme for the time-fractional Allen-Cahn equation with nondegenerate state-dependent mobility. The scheme combines a standard two-point flux approximation of diffusion, the L1 approximation of the Caputo derivative on a uniform time grid, and a semiconvex splitting of the potential, and it discretizes the inverse mobility by a secant average multiplying the entire L1 derivative. The key estimate is a lower bound for the accumulated pairing of this weighted L1 derivative with the solution increments in terms of the accumulated L1 quadratic form. Using this lower bound, we derive a discrete energy estimate and the time regularity required for compactness. For twice continuously differentiable semiconvex potentials satisfying sign conditions outside $(-1,1)$ and initial data in $H^1(\Omega)$ taking values in $[-1,1]$, we prove existence of at least one fully discrete solution on every admissible finite volume mesh and every uniform time grid. Every such solution satisfies the maximum bound principle and a discrete energy bound relative to the initial energy, without a time-step restriction. In up to three space dimensions, under a mesh regularity condition with a constant independent of the mesh size, we prove that any sequence of fully discrete solutions has a subsequence converging strongly in $L^2(\Omega\times(0,T))$ to a weak solution of the continuous problem as the mesh size and time step tend to zero. As a consequence, we obtain existence of a weak solution in one and two space dimensions, and in three dimensions whenever such meshes exist.

Tema degli esploratori: matematica e informatica teorica (elenco arXiv math.NA, nuovi lavori del giorno).
Categorie arXiv: Numerical Analysis (math.NA) ; Analysis of PDEs (math.AP).
Pagina del lavoro: https://arxiv.org/abs/2611.00837 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-11-03T18:15:22.795Z dalla pagina https://arxiv.org/list/math.NA/new (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).