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.

中文翻译：

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变密度二维纳维-斯托克斯方程能量守恒 BDF2-FE 格式的误差分析


在本文中，我们提出了变密度二维纳维-斯托克斯方程的二阶线性化有限元 (FE) 方法的误差估计。为了获得误差估计，我们首先引入原始系统的等效形式。后来，我们提出了一种通用的 BDF2-FE 方法来求解该等价形式，其中泰勒-胡德有限元空间用于离散纳维-斯托克斯方程，一致有限元空间用于离散密度方程。我们的数值格式被证明是离散级的能量耗散。在强解足够平滑的假设下，我们提出了二维变密度不可压缩流数值格式的误差估计。据我们所知，这是第一次对变密度纳维-斯托克斯方程的通用 BDF2-FE 方法（没有速度后处理）给出完整的误差估计。最后，提供了一些数值例子来证实我们的理论结果。