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

TL;DR

This paper analyzes the error of a second-order BDF2 finite element scheme for 2D variable density Navier-Stokes equations, demonstrating energy dissipation and providing theoretical error estimates with numerical validation.

Contribution

It introduces an error analysis for a BDF2-FE scheme applied to variable density Navier-Stokes equations, including energy dissipation and numerical validation.

Findings

The scheme ensures discrete energy dissipation.

Error estimates are derived under smoothness assumptions.

Numerical examples confirm theoretical results.

Abstract

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.