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.

翻译：本文针对二维变密度Navier-Stokes方程，提出了一种二阶线性化有限元方法的误差估计。为获得误差估计，我们首先引入原系统的等价形式。随后，针对该等价形式提出了一种通用的BDF2-FE方法，其中采用Taylor-Hood有限元空间离散Navier-Stokes方程，并采用协调有限元空间离散密度方程。我们证明该格式保证了离散能量耗散性。在强解充分光滑的假设下，给出了二维变密度不可压流的数值格式误差估计。最后，通过数值算例验证了理论结果。