Error analysis of energy-conservative BDF2-FE scheme for the 2D Navier-Stokes equations with variable density

In this paper, we present an error estimate of a second -order linearized finite element (FE) method for the 2D Navier-Stokes equations with variable density. In order to get error estimates, we first introduce an equivalent form of the original system. Later, we propose a general BDF2-FE method for solving this equivalent form, where the Taylor-Hood FE space is used for discretizing the Navier-Stokes equations and conforming FE space is used for discretizing density equation. Our numerical scheme is proved to be energy -dissipation in discrete level. Under the assumption of sufficient smoothness of strong solutions, an error estimate is presented for our numerical scheme for variable density incompressible flow in two dimensions. To our knowledge, this is the first time to give a complete error estimate for a general BDF2-FE method (without post -processing for velocity) for the variable density Navier-Stokes equations. Finally, some numerical examples are provided to confirm our theoretical results.