# Convergence of the MAC scheme for the incompressible Navier-Stokes equations

## TL;DR

This paper proves the convergence of the MAC scheme for incompressible Navier-Stokes equations on non-uniform grids, establishing existence, compactness, and that the limit solutions are weak solutions of the continuous problem.

## Contribution

It provides a rigorous convergence proof of the MAC scheme without regularity assumptions on solutions, extending previous results to non-uniform grids.

## Key findings

- A priori estimates on scheme solutions
- Existence of discrete solutions
- Convergence to weak solutions of Navier-Stokes

## Abstract

We prove in this paper the convergence of the Marker and cell (MAC) scheme for the dis-cretization of the steady-state and unsteady-state 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 of which tends to zero. We then establish that the limit is a weak solution to the continuous problem.