A Stable, Linear Fully Discrete Scheme and Analysis for the Cahn–Hilliard Equation With Dynamic Boundary Condition

ABSTRACT In this paper we present and analyze a stabilized linear fully discrete scheme for the Cahn–Hilliard equation with dynamic boundary condition proposed by Liu and Wu. We employ a higher order asymptotic expansion (up to second‐order temporal accuracy) and construct a correction term (higher‐order infinitesimal). This ensures that the discrete error function has zero mean, laying the foundation for its analysis in norm. By combining rough error estimates and refined error estimates, a complete convergence analysis of the scheme is accomplished. Specifically, by treating the numerical solution as a small perturbation of the exact solution, we prove the uniform boundedness of the numerical solution and its temporal difference in the discrete norm based on the results of the rough error estimate. Subsequently, the refined error estimate is derived to obtain the optimal rate of convergence, with the help of bound of the numerical solution and its temporal difference. Our stability analysis requires neither the global Lipschitz assumption nor cut‐off for the nonlinear term, which significantly improves the corresponding findings in the previous literature. Some numerical experiments are performed to verify the optimal convergence order and the robustness of the proposed scheme.