A penalty finite volume method for the transient Navier–Stokes equations

A fully discrete penalty finite volume method is introduced for the discretization of the two-dimensional transient Navier–Stokes equations, where the temporal discretization is based on a backward Euler scheme and the spatial discretization is based on a finite volume scheme that uses a pair of P2–P0 trial functions on triangles. This method allows us to efficiently separate the computation of velocity from that of pressure with reasonably large time steps, and conserves mass locally. In addition, error estimates of optimal order are obtained for the fully discrete method under reasonable assumptions on temporal and spatial step sizes and the physical data. Finally, we present two numerical examples to illustrate the numerical algorithms developed and to show numerical results that agree with the theory established.