A hybrid FEM for solving the Allen–Cahn equation

• We use the hybrid method for solving the Allen–Cahn equation.
• The numerical method is based on the finite element method.
• We simulate the evolution of the Allen–Cahn equation on the complex domains.

We present an unconditionally stable hybrid finite element method for solving the Allen–Cahn equation, which describes the temporal evolution of a non-conserved phase-field during the antiphase domain coarsening in a binary mixture. Its various modified forms have been applied to image analysis, motion by mean curvature, crystal growth, topology optimization, and two-phase fluid flows. The hybrid method is based on the operator splitting method. The equation is split into a heat equation and a nonlinear equation. An implicit finite element method is applied to solve the diffusion equation and then the nonlinear equation is solved analytically. Various numerical experiments are presented to confirm the accuracy and efficiency of the method. Our simulation results are consistent with previous theoretical and numerical results.