  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. We show that our scheme ensures discrete energy dissipation.
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. Finally, some numerical examples are
provided to confirm our theoretical results.

Numerical Analysis;Numerical Analysis