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 taken into account in the construction. The interface is approximated using discrete level set functions, and explicit formulas for IFE basis functions and correction functions are derived, facilitating ease of implementation.The inf-sup stability and the optimal a priori error estimate of the IFE method, along with the optimal approximation capabilities of the IFE space, are derived rigorously, with constants that are independent of the mesh size and the manner in which the interface intersects the mesh, but may depend on the discontinuous viscosity coefficients. Additionally, it is proved that the condition number has the usual bound independent of the interface. Numerical experiments are provided to confirm the theoretical results.

中文翻译：

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笛卡尔网格上两相斯托克斯问题的微型浸入有限元法


本文提出了一种用于求解笛卡尔网格上二维和三维两相斯托克斯问题的微型浸入式有限元 (IFE) 方法。 IFE空间由传统的迷你单元构建，根据界面跳跃条件修改界面单元的形状函数，同时保持自由度不变。构造中考虑了不连续粘度系数和表面力。使用离散水平集函数对接口进行近似，并推导了IFE基函数和校正函数的显式公式，便于实现。IFE方法的inf-sup稳定性和最优先验误差估计以及最优近似IFE 空间的能力是严格导出的，其常数与网格尺寸和界面与网格相交的方式无关，但可能取决于不连续粘度系数。此外，还证明了条件数具有与接口无关的通常界限。提供数值实验来证实理论结果。