A structure-preserving finite volume scheme for the time-fractional Allen-Cahn equation with nondegenerate state-dependent mobility
Abstract
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 H1(Ω) 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 L2(Ω×(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.
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BibTeX
@misc{jeon2026structurepreserving,
title = {{A structure-preserving finite volume scheme for the time-fractional Allen-Cahn equation with nondegenerate state-dependent mobility}},
author = {Yonghyeon Jeon and Kiwoong Kwon},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.00837},
eprint = {2611.00837},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.00837},
}
RIS
TY - UNPB
TI - A structure-preserving finite volume scheme for the time-fractional Allen-Cahn equation with nondegenerate state-dependent mobility
AU - Yonghyeon Jeon
AU - Kiwoong Kwon
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.00837
UR - https://arxiv.org/abs/2611.00837
AB - 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(Ω) 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(Ω×(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.
ER -