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:虽然计算方法产生准确的解决方案,并且可以通过分辨率研究来证明数字趋同,但严格的趋同证明通常仅限于特定的重整和边界条件。在本文中,对斯托克斯的标记和细胞(MAC)办法进行了严格的错误分析,在有限差异方法的框架内,对斯托克斯的标记和细胞(MAC)办法的接口问题与固定的粘度问题进行了严格的分析。在不将问题重新纳入椭圆式PDE的情况下,主要的想法是使用离散的Ladyzenskaja-Bususka-Brezzi(LBB)条件和构建辅助功能,这些功能能满足离散式斯托克斯方程式的特性,并在移动界面附近至少具有二级精确性。特别是,这种方法首次使离散的 $\ell=2美元-诺尔姆的速度梯度能证明二次相趋联,除了速度和压力场外,速度梯度的加速梯度的二次相趋同。数字实验核查方法的预期特性以及二维和三维示例的预期精确度。