Numerical analysis of fully discrete energy stable weak Galerkin finite element Scheme for a coupled Cahn-Hilliard-Navier-Stokes phase-field model

The Cahn-Hilliard phase-field model of two-phase incompressible flows, namely the Cahn-Hilliard-Navier-Stokes (CH-NS) problem represents the fundamental building blocks of hydrodynamic phase-field models for multiphase fluid flow dynamics. Since finding solution (numerically and theoretically) of the CH-NS system is non-trivial (owing to the coupling between the Navier-Stokes equation and the Cahn-Hilliard equation), in this paper, we propose and analyze a numerical scheme with the following properties: (1) first-order in time, (2) nonlinear, (3) fully coupled and (4) energy stable; for solving CH-NS system, in the framework of weak Galerkin (WG) method. More precisely, we employ the WG method, which uses discontinuous functions to construct the approximation space, and the first-order backward Euler (implicit) method for space and time discretizations, respectively. We first recall the corresponding variational formulation, and then summarize the main WG method ingredients that are required for our discrete analysis. In particular, in order to define the weak discrete bilinear (and trilinear) form, whose continuous version involves classical differential operators, we propose two well-known alternatives for gradient and divergence operators onto a suitable polynomial subspace. Next, we show that the weak global discrete bilinear form satisfies the hypotheses required by the Lax-Milgram lemma. In this way, we derive the associated a priori error estimates for phase field variable, chemical potential, velocity and further the pressure in L2 norm. Finally, several numerical results confirming the theoretical rates of convergence and illustrating the good performance of the method for simulation influence of surface tension, small density variations, and the driven cavity are presented.