This paper presents a mini immersed finite element (IFE) method for solving two- and three-dimensional two-phase Stokes problems on Cartesian meshes. The IFE space is constructed from the conventional mini element with shape functions modified on interface elements according to interface jump conditions, while keeping the degrees of freedom unchanged. Both discontinuous viscosity coefficients and surface forces are considered in the construction. The interface is approximated via discrete level set functions and explicit formulas of IFE basis functions and correction functions are derived, which make the IFE method easy to implement. The optimal approximation capabilities of the IFE space and the inf-sup stability and the optimal a priori error estimate of the IFE method are derived rigorously with constants independent of the mesh size and how the interface cuts the mesh. It is also proved that the condition number has the usual bound independent of the interface. Numerical experiments are provided to confirm the theoretical results.

翻译：本文提出了一种用于在笛卡尔网格上求解二维和三维两相Stokes问题的微型浸入式有限元（IFE）方法。该IFE空间基于传统微型单元构建，其形函数在界面单元上根据界面跳跃条件进行修正，同时保持自由度不变。在构建过程中考虑了不连续的黏性系数和表面力。界面通过离散水平集函数近似，并推导了IFE基函数和修正函数的显式公式，使得该方法易于实现。本文严格论证了IFE空间的最优逼近能力，以及该方法的inf-sup稳定性和最优先验误差估计，其中常数与网格尺寸及界面切割网格的方式无关。同时证明了条件数具有与界面无关的常规上界。数值实验验证了理论结果。