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.

翻译：Stokes流动方程已成功应用于模拟具有移动界面的实际问题。尽管计算方法能够生成精确解，且可通过分辨率研究证明数值收敛性，但严格的收敛性证明通常局限于特定的重构形式和边界条件。本文在有限差分方法框架下，针对具有常黏度的Stokes界面问题，对标记点与网格（MAC）格式进行了严格的误差分析。无需将问题重构为椭圆型偏微分方程，其主要思想是利用离散Ladyzenskaja-Babuska-Brezzi（LBB）条件并构造辅助函数，这些函数满足离散Stokes方程，且在移动界面附近至少具有二阶精度。特别地，该方法首次使得在离散ℓ²范数下，除速度和压力场外，还能证明速度梯度的二阶收敛性。数值实验验证了该方法在二维和三维算例中的理想性质及预期精度阶数。