A Convergent Staggered Scheme for the Variable Density Incompressible Navier-Stokes Equations

TL;DR

This paper introduces a convergent staggered finite element scheme for the variable density Navier-Stokes equations, ensuring stability and convergence to weak solutions through energy estimates and compactness arguments.

Contribution

It develops a novel implicit scheme combining finite element and finite volume methods with proven stability and convergence for variable density incompressible flows.

Findings

Scheme preserves stability properties of the continuous problem

Solutions converge to weak solutions as discretization parameters tend to zero

Method ensures discrete kinetic energy balance and density bounds

Abstract

In this paper, we analyze a scheme for the time-dependent variable density Navier-Stokes equations. The algorithm is implicit in time, and the space approximation is based on a low-order staggered non-conforming finite element, the so-called Rannacher-Turek element. The convection term in the momentum balance equation is discretized by a finite volume technique, in such a way that a solution obeys a discrete kinetic energy balance, and the mass balance is approximated by an upwind finite volume method. We first show that the scheme preserves the stability properties of the continuous problem (L $\infty$-estimate for the density, L $\infty$ (L 2)-and L 2 (H 1)-estimates for the velocity), which yields, by a topological degree technique, the existence of a solution. Then, invoking compactness arguments and passing to the limit in the scheme, we prove that any sequence of solutions (obtained with a sequence of discretizations the space and time step of which tend to zero) converges up to the extraction of a subsequence to a weak solution of the continuous problem.