Error analysis of energy-conservative BDF2-FE scheme for the 2D Navier-Stokes equations with variable density (Q6591756)

The authors design a BDF2-FEM scheme in order to numerically solve an initial-boundary value problem for the 2D incompressible Navier-Stokes system involving variable density. They use the Taylor-Hood (P2-P1) finite element space for discretising velocity \(u\) and pressure \(p\), along with the conforming finite element (P2) for density variable. The unconditional energy stability of the BDF2 coupled with this scheme for spatial discretisation is proved. In the definition of discrete energy, however, only the discrete velocity and the discrete density enter, but not the discrete pressure. The proof of this fundamental result occupies almost nine pages of a huge number of successive inequalities! Three numerical examples are carried out to underline the capabilities of the proposed algorithm. They refer to some 2D convex domains.