A linear, fully decoupled, and unconditionally energy-stable SAV-FEM for the Cahn--Hilliard--Navier--Stokes model

In this paper, we develop a linear, fully decoupled, and unconditionally energy-stable fully discrete finite element scheme for the Cahn--Hilliard--Navier--Stokes (CHNS) system by employing the scalar auxiliary variable (SAV) approach.

Unlike existing SAV-based formulations that typically introduce multiple auxiliary variables or additional techniques to handle different nonlinearities, we introduce only one scalar auxiliary variable together with a novel update of the auxiliary variable to reformulate all nonlinear terms arising from the Cahn--Hilliard and Navier--Stokes equations, yielding an equivalent reformulation of the original CHNS system.

An implicit--explicit (IMEX) Euler scheme is applied for temporal discretization, where the linear terms are treated implicitly and the nonlinear terms explicitly, while a finite element method is adopted for spatial discretization.

The resulting fully discrete scheme can be efficiently decomposed into two linear subproblems and one scalar quadratic algebraic equation, which significantly simplifies the implementation.

Furthermore, we prove that the proposed scheme satisfies an unconditional discrete energy dissipation law and establish its stability with respect to several relevant norms.

Optimal-order $L^2$ error estimates are also derived for the fully discrete finite element approximation.

Finally, a series of numerical experiments are presented to verify the theoretical results and demonstrate the efficiency of the proposed method.