The Allen-Cahn equation with a time Caputo-Hadamard derivative: Mathematical and Numerical Analysis

Abstract

<abstract><p>In this paper, we investigate the local discontinuous Galerkin (LDG) finite element method for the fractional Allen-Cahn equation with Caputo-Hadamard derivative in the time domain. First, the regularity of the solution is analyzed, and the results indicate that the solution of this equation generally possesses initial weak regularity in the time dimension. Due to this property, a logarithmic nonuniform L1 scheme is adopted to approximate the Caputo-Hadamard derivative, while the LDG method is used for spatial discretization. The stability and convergence of this fully discrete scheme are proven using a discrete fractional Gronwall inequality. Numerical examples demonstrate the effectiveness of this method.</p></abstract>