Two-Grid Finite Element Method for the Time-Fractional Allen-Cahn Equation With the Logarithmic Potential

In this paper, we propose a two-grid finite element method for solving the time-fractional Allen-Cahn equation with the logarithmic potential. Firstly, with the L1 method to approximate Caputo fractional derivative, we solve the fully discrete time-fractional Allen-Cahn equation on a coarse grid with mesh size H$$ H $$ and time step size tau$$ \tau $$. Then, we solve the linearized system with the nonlinear term replaced by the value of the first step on a fine grid with mesh size h$$ h $$ and the same time step size tau$$ \tau $$. We obtain the energy stability of the two-grid finite element method and the optimal order of convergence of the two-grid finite element method in the L2 norm when the mesh size satisfies h=O(H2)$$ h=O\left({H}<^>2\right) $$. The theoretical results are confirmed by arithmetic examples, which indicate that the two-grid finite element method can keep the same convergence rate and save the CPU time.