The numerical solution of the Stokes equations on an evolving domain with a moving boundary is studied based on the arbitrary Lagrangian-Eulerian finite element method and a second-order projection method along the trajectories of the evolving mesh for decoupling the unknown solutions of velocity and pressure. The error of the semidiscrete arbitrary Lagrangian-Eulerian method is shown to be $O(h^{r+1})$ for the Taylor--Hood finite elements of degree $r\ge 2$, using Nitsche's duality argument adapted to an evolving mesh, by proving that the material derivative and the Stokes--Ritz projection commute up to terms which have optimal-order convergence in the $L^2$ norm. Additionally, the error of the fully discrete finite element method, with a second-order projection method along the trajectories of the evolving mesh, is shown to be $O(\ln(1/\tau+1)\tau^{2}+\ln(1/h+1)h^{r+1})$ in the discrete $L^\infty(0,T; L^2)$ norm using newly developed energy techniques and backward parabolic duality arguments that are applicable to the Stokes equations with an evolving mesh. To maintain consistency between the notations of the numerical scheme in a moving domain and those in a fixed domain, we introduce the equivalence class of finite element spaces across time levels. Numerical examples are provided to support the theoretical analysis and to illustrate the performance of the method in simulating Navier--Stokes flow in a domain with a rotating propeller.

翻译：针对移动边界演化域上的 Stokes 方程数值求解，基于任意拉格朗日-欧拉有限元方法与沿演化网格轨迹的二阶投影方法解耦未知速度与压力解。通过将 Nitsche 对偶论证适配至演化网格，证明物质导数与 Stokes-Ritz 投影在 $L^2$ 范数意义下交换至最优阶收敛项，进而得到半离散任意拉格朗日-欧拉方法对 $r\ge 2$ 阶 Taylor-Hood 有限元具有 $O(h^{r+1})$ 误差。进一步，利用针对演化网格 Stokes 方程的新能量技术与逆向抛物对偶论证，证明全离散有限元方法（结合沿演化网格轨迹的二阶投影方法）在离散 $L^\infty(0,T; L^2)$ 范数下误差为 $O(\ln(1/\tau+1)\tau^{2}+\ln(1/h+1)h^{r+1})$。为统一移动域与固定域中数值格式的符号体系，引入跨时间层的有限元空间等价类。数值算例验证理论分析，并展示该方法在模拟含旋转叶轮域内 Navier-Stokes 流动中的性能。