Convergence analysis of a decoupled pressure-correction SAV-FEM for the Cahn–Hilliard–Navier–Stokes model

Abstract

In this paper, a linear, decoupled and unconditional energy-stable numerical scheme of Cahn–Hilliard–Navier–Stokes (CHNS) model is constructed and analyzed. We reformulate the CHNS model into an equivalent system based on the scalar auxiliary variable approach. The Euler implicit/explicit scheme is used for time discretization, that is, linear terms are implicitly processed, nonlinear terms are explicitly processed, and pressure-correction method of the Navier–Stokes equations is also adopted. In this way, we achieve the decoupling of phase field, velocity and pressure, and only need to solve a series of linear equations with constant coefficients at each time step, thereby improving computational efficiency. Then the finite element method is used for space discretization, and the finite element spaces of phase field, velocity and pressure are taken as Pl−Pl−Pl−1 respectively. We also verify that the proposed scheme satisfies the discrete energy dissipation law. The optimal error estimates of the fully discrete scheme is proved, which the phase field, velocity and pressure satisfy the first-order accuracy in time and the (ℓ+1,ℓ+1,ℓ)th order accuracy in space, respectively. Finally, numerical examples are presented to demonstrate the accuracy and efficiency of the proposed method.