Numerical Analysis of Stabilized Second Order Semi-Implicit Finite Element Methods for the Phase-Field Equations

In this paper, we consider two stabilized second-order semi-implicit finite element methods for solving the Allen-Cahn and Cahn-Hilliard equations. Stabilized semi-implicit schemes are used for temporal discretization, and the finite element method is used for spatial discretization. It is shown that by adding a single linear term that is of the same order with the truncation error in time, the proposed methods are all unconditionally energy stable. Error estimates for the two schemes are also established. Numerical examples are presented to confirm the accuracy, efficiency and stability of the proposed methods.