Stabilized finite element methods based on multiscale enrichment for Allen-Cahn and Cahn-Hilliard equations
Juan Wen, Ya‐Ling He, Yinnian He, Kun Wang · Communications on Pure & Applied Analysis · 2021
In this paper, we investigate fully discrete schemes for the Allen-Cahn and Cahn-Hilliard equations respectively, which consist of the stabilized finite element method based on multiscale enrichment for the spatial discretization and the semi-implicit scheme for the temporal discretization. With reasonable stability conditions, it is shown that the proposed schemes are energy stable. Furthermore, by defining a new projection operator, we deduce the optimal \begin{document}$ L^2 $\end{document} error estimates. Some numerical experiments are presented to confirm the theoretical predictions and the efficiency of the proposed schemes.