Fully-discrete finite element numerical scheme with decoupling structure and energy stability for the Cahn–Hilliard phase-field model of two-phase incompressible flow system with variable density and viscosity

We construct a fully-discrete finite element numerical scheme for the Cahn–Hilliard phase-field model of two-phase incompressible flow system with variable density and viscosity. The is linear, decoupled, unconditionally energy stable. Its key idea to combine penalty method Navier–Stokes equations Strang operator splitting method, introduce several nonlocal variables their ordinary differential process coupled nonlinear terms. highly efficient it only needs solve series completely independent linear elliptic at each time step, in which equation pressure Poisson have constant coefficients. rigorously prove unconditional stability solvability carry out numerous accuracy/stability examples various benchmark simulations 2D 3D, including Rayleigh–Taylor instability rising/coalescence dynamics bubbles demonstrate effectiveness scheme, numerically.