A two-grid finite element method for the Allen-Cahn equation with the logarithmic potential

In this paper, we present a two-grid finite element method for the Allen-Cahn equation with the logarithmic potential. This method consists of two steps. In the first step, based on a fully implicit finite element method, the Allen-Cahn equation is solved on a coarse grid with mesh size H. In the second step, a linearized system whose nonlinear term is replaced by the value of the first step is solved on a fine grid with mesh size h. We give the energy stabilities of the traditional finite element method and the two-grid finite element method. The optimal convergence order of the two-grid finite element method in H-1 norm is achieved when the mesh sizes satisfy h = O(H-2). Numerical examples are given to demonstrate the validity of the proposed scheme. The results show that the two-grid method can save the CPU time while keeping the same convergence rate.

Automated summary

In this paper, a two-grid finite element method for the Allen-Cahn equation with the logarithmic potential is presented. The method consists of two steps, solving the equation on coarse and fine grids respectively. The energy stabilities and convergence properties of the method are discussed and validated through numerical examples.