Two Unconditionally Energy Stable Schemes for the Cahn-Hilliard Equation

Abstract

In this paper, we propose two fully discrete convex splitting schemes to solve the Cahn-Hilliard equation based on the mixed finite element method. For these two numerical schemes, the first-order backward Euler method and the second-order backward differentiation formula (BDF2) are used for temporal discretization, and the nonlinear term is treated by the convex splitting method. In order to ensure unconditional energy stability, the second-order time scheme requires the addition of a stability term [[EQUATION]], where [[EQUATION]] is a stable parameter. We strictly prove that both numerical schemes have unconditional energy stability. In particular, the second-order time scheme can be guaranteed to be unconditionally energy stable for [[EQUATION]]. Additionally, we conduct rigorous error analysis on these two numerical schemes and obtain optimal error estimates in [[EQUATION]] norm. Lastly, we verify the effectiveness of both numerical schemes and confirm the correctness of the theoretical results.

Keywords: Cahn-Hilliard equation, Mixed finite element method, Convex splitting method, unconditional energy stability, Error estimate.