Stability and superconvergence of MAC schemes for time dependent Stokes equations on nonuniform grids

In this paper, two MAC schemes are introduced and analyzed to solve the time dependent Stokes equations on nonuniform grids. One scheme is the Euler backward scheme with first order accuracy in time increment while the other one is the Crank Nicolson scheme with second order accuracy in time increment. By constructing an auxiliary function depending on the velocity and discretizing parameters, we obtain the second order superconvergence in L2 norm for both velocity and pressure. Besides, second order superconvergence for some terms of H1 norm of the velocity on nonuniform grids is obtained. Finally, some numerical experiments are presented to show that the convergence rates are in agreement with the theoretical analysis.