Here, we carry out rigorous error analysis for a first-order in time, linear, fully decoupled and energy stable scheme for solving the Cahn–Hilliard phase-field model of two-phase incompressible flows, namely Cahn–Hilliard–Navier–Stokes problem (Shen and Yang, SIAM J Numer Anal, 2015). The error estimates are for phase field variable, chemical potential, velocity and further the pressure in \(L^2\) norm and \(L^{\infty }\) norm. The scheme combines the projection method, the explicit stabilizing decoupling technique, and the linear stabilization approach together. We further derive the boundness of numerical solution in \(L^\infty \) norm with the mathematical deduction, and deal with the complex splitting error arising from the decoupling technique. Optimal error estimates are derived for the semi-discrete-in-time scheme.

在这里，我们对一阶时间，线性，完全解耦且能量稳定的方案进行了严格的误差分析，以解决两相不可压缩流的Cahn-Hilliard相场模型，即Cahn-Hilliard-Navier-Stokes问题（Shen and Yang，SIAM J Numer Anal，2015年）。误差估计是针对相场变量，化学势，速度以及\（L ^ 2 \）规范和\（L ^ {\ infty} \）规范中的压力。该方案将投影方法，显式稳定解耦技术和线性稳定方法结合在一起。我们进一步导出\（L ^ \ infty \）中数值解的界用数学推导进行归一化，并处理因去耦技术而产生的复杂分裂误差。最佳误差估计是针对半离散时间方案得出的。