A two‐grid finite element method for the Allen‐Cahn equation with the logarithmic potential
Danxia Wang, Yanan Li, Hongen Jia
Abstract
Abstract 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.