Second-Order Error Analysis for Fractal Mobile/Immobile Allen-Cahn Equation on Graded Meshes

The fractal mobile/immobile model bridges between Fickian fluxes at early times and nonGaussian behavior at late times. In this work, an averaged L1 scheme for solving the fractal mobile/immobile Allen-Cahn equation with a Caputo temporal derivative of order alpha is an element of (0, 1) is developed and analyzed on graded meshes. The unique solvability and discrete energy stability are established rigorously on arbitrary nonuniform time meshes. Based on the spectral norm inequality, the unconditional stability and the second-order convergence analysis under the weakly regularity assumption are investigated on graded meshes. Finally, several numerical examples are presented to illustrate the theoretical analysis. To the best of our knowledge, this is the first topic on the convergence analysis for the fractal mobile/immobile Allen-Cahn equation on graded meshes.