AI 中文总结
本文针对非拟合网格的Stokes界面问题，提出一种压力鲁棒非协调浸入有限元方法，通过分解压力分量和重构测试函数实现压力鲁棒性，经数值实验验证了理论结果。
AI 中文摘要
众所周知，对测试函数进行适当修改可得到Stokes问题的压力鲁棒混合方法。但对于非拟合网格上Stokes界面问题的浸入有限元近似，由于速度和压力在某一界面条件中耦合，速度误差是否与压力无关仍不明确。本文通过将不连续压力分解为连续分量和依赖于速度的不连续分量，给出了肯定答案。我们证明，浸入Crouzeix--Raviart/$P_0$单元方法通过在右端项上对测试函数进行$H(\text{div})$协调重构，实现了压力鲁棒性。建立了所提方法的稳定性和最优误差估计，其常数与界面相对于网格的位置无关。数值实验验证了理论结果。
英文摘要
It is well established that an appropriate modification of test functions may lead to pressure-robust mixed methods for Stokes problems. However, for immersed finite element approximations of Stokes interface problems on unfitted meshes, it remains unclear whether the velocity error is independent of the pressure, since the velocity and the pressure are coupled in one of the interface conditions. In this paper, we provide a positive answer through a novel decomposition of the discontinuous pressure into a continuous component and a velocity-dependent discontinuous component. We demonstrate that the immersed Crouzeix--Raviart/$P_0$ element method achieves pressure robustness via an $H(\operatorname{div})$-conforming reconstruction of the test functions on the right-hand side. The stability and optimal error estimates of the proposed method are established with constants independent of the interface position relative to the mesh. Numerical experiments are presented to validate the theoretical findings.
