A Fully-Discrete Decoupled Finite Element Method for the Conserved Allen-Cahn Type Phase-Field Model of Three-Phase Fluid Flow System

Xiaofeng Yang
Xiaoming He, Missouri University of Science and Technology

He’s research is partially supported by the U.S. National Science Foundation under grant number DMS-1818642.


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 L2-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.