In this article, we develop and analyze a novel fully discrete decoupled finite element method to solve a flow-coupled ternary phase-field model for the system consisting of three immiscible fluid components. Based on the $L^2$-gradient flow approach, the conserved Allen–Cahn type dynamics is used to describe the free interface motion, where multiple nonlocal type Lagrange multipliers are used to accurately conserve the volume of each phase. The scheme is also linear, second-order time accurate, and unconditionally energy stable, due to the combination of several effective numerical techniques, including the two-step backward differentiation scheme, finite element discretization, explicit-SAV (scalar auxiliary variable) method for handling the nonlinearity, and projection method of Navier–Stokes equation. At each time step, the non-local splitting technique only requires solving several decoupled constant-coefficient elliptic equations. The implementation issues are discussed in detail. The solvability and the unconditional energy stability of the scheme are rigorously proved. Plenty of 2D and 3D numerical simulations are carried out to numerically demonstrate the accuracy, energy stability, and applicability of the proposed scheme.

在本文中，我们开发并分析了一种新颖的完全离散解耦有限元方法，以求解由三个不混溶流体成分组成的系统的流动耦合三元相场模型。基于${升}^2$-梯度流方法，使用守恒的 Allen-Cahn 型动力学来描述自由界面运动，其中使用多个非局部型拉格朗日乘子来精确守恒每相的体积。由于结合了几种有效的数值技术，包括两步向后微分方案、有限元离散化、显式 SAV（标量辅助变量）方法处理非线性和纳维-斯托克斯方程的投影方法。在每个时间步，非局部分裂技术只需要求解几个解耦的常数系数椭圆方程。详细讨论了实施问题。严格证明了该方案的可解性和无条件能量稳定性。进行了大量的 2D 和 3D 数值模拟，以数值方式证明所提出方案的准确性、能量稳定性和适用性。