A Convergent Staggered Scheme for the Variable Density Incompressible NavierStokes Equations
Abstract
In this paper, we analyze a scheme for the timedependent variable density NavierStokes equations. The algorithm is implicit in time, and the space approximation is based on a loworder staggered nonconforming finite element, the socalled RannacherTurek 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.
 Publication:

arXiv eprints
 Pub Date:
 March 2016
 arXiv:
 arXiv:1603.07221
 Bibcode:
 2016arXiv160307221L
 Keywords:

 Mathematics  Numerical Analysis;
 Mathematics  Analysis of PDEs;
 Physics  Classical Physics