Convergence of the Marker-and-Cell Scheme for the Incompressible Navier–Stokes Equations on Non-uniform Grids

We prove in this paper the convergence of the marker-and-cell scheme for the discretization of the steady-state and time-dependent incompressible navier–stokes equations in primitive variables, on non-uniform cartesian grids, without any regularity assumption on the solution. a priori estimates on solutions to the scheme are proven; they yield the existence of discrete solutions and the compactness of sequences of solutions obtained with family of meshes the space step and, for the time-dependent case, the time step of which tend to zero. we then establish that the limit is a weak solution to the continuous problem.