A convergent staggered scheme for the variable density incompressible Navier-Stokes equations

Source 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 ∞ \mathrm {L}^\infty -estimate for the density, L ∞ ( L 2 ) \mathrm {L}^\infty (\mathrm {L}^2) - and L 2 ( H 1 ) \mathrm {L}^2(\mathrm {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.