Maximum error estimates of a MAC scheme for Stokes equations with Dirichlet boundary conditions

Abstract

This paper presents a rigorous convergence analysis in ℓ2-norm and maximum norm for the Marker-And-Cell (MAC) scheme for solving the incompressible Stokes equations with Dirichlet boundary conditions in two dimensions. The numerical scheme is based on a finite-difference approximation on a staggered grid, where the computational boundary conditions are introduced by reflecting boundary conditions. This scheme is well known to numerically produce second order accuracy for velocity, pressure as well as the gradient of velocity. However, the reflecting boundary conditions lead to the fact that the truncation errors near the boundary is the order of O(1), which presents great difficulties in error analysis to obtain optimal order. In this work, by means of a discrete LBB condition and some auxiliary functions that satisfy discrete Stokes equations to a high-order accuracy, we first prove that the convergence is second order in the ℓ2-norm for velocity, pressure and the gradient of the velocity. After that, according to the property of discrete Green functions, we further provide rigorous analysis to show second-order accuracy in the maximum norm for both the velocity and its gradient. The extension of our results to three dimensional problems would be routine. Numerical results are also provided to demonstrate our theoretical analysis.