Performance of a new stabilized structure-preserving finite element method for the Allen–Cahn equation

Abstract

This paper presents an energy stable scheme for the numerical solution of the Allen–Cahn equation. The scheme combines the Crank–Nicolson/Adams–Bashforth technique with the Galerkin method, complementing a thoughtfully designed stabilization term. While the Allen–Cahn equation is inherently nonlinear, our proposed algorithm effectively addresses this nonlinearity by utilizing the implicit nature of the resulting system of equations. Additionally, the [Formula: see text] error estimate for the fully discretized scheme is carried out in a rigorous way with the optimal error bound [Formula: see text]. This analysis demonstrates the precision and accuracy of our approach. In tandem with the theoretical framework, we conduct comprehensive numerical experiments to assess the practical validity of our method. Our findings not only validate the theoretical results but also reveal an intriguing observation: the judiciously chosen stabilization term significantly enhances the performance of the model. This finding underscores the practical effectiveness of our proposed scheme, which consistently produces precise and reliable solutions, reaffirming existing evidence.

Keywords

• Finite element method
• Crank–Nicolson/Adams–Bashforth
• Allen–Cahn equation
• error estimates