For decades, the widely used finite difference method on staggered grids, also
known as the marker and cell (MAC) method, has been one of the simplest and most effective
numerical schemes for solving the Stokes equations and Navier–Stokes equations. Its
superconvergence on uniform meshes has been observed by Nicolaides (SIAM J Numer Anal
29(6):1579–1591, 1992), but the rigorous proof is never given. Its behavior on non-uniform
grids is not well studied, since most publications only consider uniform grids. In this work,
we develop the MAC scheme on non-uniform rectangular meshes, and for the first time we
theoretically prove that the superconvergence phenomenon (i.e., second order convergence in
the L2 norm for both velocity and pressure) holds true for the MAC method on non-uniform
rectangular meshes. With a careful and accurate analysis of various sources of errors, we
observe that even though the local truncation errors are only first order in terms of mesh
size, the global errors after summation are second order due to the amazing cancellation
of local errors. This observation leads to the elegant superconvergence analysis even with
non-uniform meshes. Numerical results are given to verify our theoretical analysis.