  Stokes flow equations have been implemented successfully in practice for
simulating problems with moving interfaces. Though computational methods
produce accurate solutions and numerical convergence can be demonstrated using
a resolution study, the rigorous convergence proofs are usually limited to
particular reformulations and boundary conditions. In this paper, a rigorous
error analysis of the marker and cell (MAC) scheme for Stokes interface
problems with constant viscosity in the framework of the finite difference
method is presented. Without reformulating the problem into elliptic PDEs, the
main idea is to use a discrete Ladyzenskaja-Babuska-Brezzi (LBB) condition and
construct auxiliary functions, which satisfy discretized Stokes equations and
possess at least second order accuracy in the neighborhood of the moving
interface. In particular, the method, for the first time, enables one to prove
second order convergence of the velocity gradient in the discrete
$\ell^2$-norm, in addition to the velocity and pressure fields. Numerical
experiments verify the desired properties of the methods and the expected order
of accuracy for both two-dimensional and three-dimensional examples.

Numerical Analysis;Numerical Analysis