A highly efficient operator-splitting finite element method for 2D/3D nonlinear Allen-Cahn equation

Purpose - This study aims to present a highly efficient operator-splitting finite element method for the nonlinear two-dimensional/three-dimensional (2D/3D) Allen-Cahn (AC) equation which describes the anti-phase domain coarsening in a binary alloy. This method is presented to overcome the higher storage requirements, computational complexity and the nonlinear term in numerical computation for the 2D/3D AC equation. Design/methodology/approach - In each time interval, the authors first split the original equation into a heat equation and a nonlinear equation. Then, they split the high-dimensional heat equation into a series of one-dimensional (1D) heat equations. By solving each 1D subproblem, the authors obtain a numerical solution for heat equation and take it as an initial for the nonlinear equation, which is solved analytically. Findings - The authors show that the proposed method is unconditionally stable. Finally, various numerical experiments are presented to confirm the high accuracy and efficiency of this method. Originality/value - A new operator-splitting method is presented for solving the 2D/3D parabolic equation. The 2D/3D parabolic equation is split into a sequence of 1D parabolic equations. In comparison with standard finite element method, the present method can save much central processing unit time. Stability analysis and error estimates are derived and numerical results are presented to support the theoretical analysis.