STABILITY AND SUPERCONVERGENCE OF MAC SCHEME FOR STOKES EQUATIONS ON NONUNIFORM GRIDS

摘要

The marker and cell (MAC) method, a class of finite volume schemes based on staggered grids, has been one of the simplest and most effective numerical schemes for solving the Stokes and Navier Stokes equations. Its numerical superconvergence on uniform grids has been observed since 1992. In this paper we establish the LBB condition and the stability for both velocity and pressure for the MAC scheme of stationary Stokes equations on nonuniform grids. Then we construct an auxiliary function depending on the velocity and discretizing parameters and analyze the superconvengence. We obtain the second order superconvergence in the L2 norm for both velocity and pressure for the MAC scheme. We also obtain the second order superconvergence for some terms of the H1 norm of the velocity, and the other terms of the H1 norm are second order superconvergent on uniform grids. Our analysis can be extended to three dimensional problems. Numerical experiments using the MAC scheme show agreement of the numerical results with theoretical analysis.