This paper addresses the analysis and numerical assessment of a computational method for solving the Cahn--Hilliard equation defined on a surface. The proposed approach combines the stabilized trace finite element method for spatial discretization with an implicit--explicit scheme for temporal discretization. The method belongs to a class of unfitted finite element methods that use a fixed background mesh and a level-set function for implicit surface representation. We establish the numerical stability of the discrete problem by showing a suitable energy dissipation law for it. We further derive optimal-order error estimates assuming simplicial background meshes and finite element spaces of order $m \geq 1$. The effectiveness of the method is demonstrated through numerical experiments on several two-dimensional closed surfaces, confirming the theoretical results and illustrating the robustness and convergence properties of the scheme.

翻译：本文针对定义在曲面上的Cahn-Hilliard方程求解方法进行了理论分析与数值评估。所提出的方法将用于空间离散化的稳定迹有限元方法与用于时间离散化的隐式-显式格式相结合。该方法属于一类非拟合有限元方法，其使用固定的背景网格和水平集函数进行隐式曲面表示。我们通过证明离散问题满足适当的能量耗散定律，确立了其数值稳定性。进一步地，在假设使用单纯形背景网格和阶数 $m \geq 1$ 的有限元空间条件下，我们推导了最优阶误差估计。通过在多个二维闭合曲面上的数值实验，验证了该方法的有效性，结果证实了理论分析，并展示了该方案的鲁棒性与收敛性。