A structure-preserving finite volume scheme for the time-fractional Allen-Cahn equation with nondegenerate state-dependent mobility
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 $L^2(Ω\times(0,T))$ 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.