In this paper, we develop a new immersed finite element method (IFEM) for two-phase incompressible Stokes flows. We allow the interface to cut the finite elements. On the noninterface element, the standard Crouzeix–Raviart element and the P 0 {P_{0}} element pair is used. On the interface element, the basis functions developed for scalar interface problems (Kwak et al., An analysis of a broken P 1 {P_{1}} -nonconforming finite element method for interface problems, SIAM J. Numer. Anal. (2010)) are modified in such a way that the coupling between the velocity and pressure variable is different. There are two kinds of basis functions. The first kind of basis satisfies the Laplace–Young condition under the assumption of the continuity of the pressure variable. In the second kind, the velocity is of bubble type and is coupled with the discontinuous pressure, still satisfying the Laplace–Young condition. We remark that in the second kind the pressure variable has two degrees of freedom on each interface element. Therefore, our methods can handle the discontinuous pressure case. Numerical results including the case of the discontinuous pressure variable are provided. We see optimal convergence orders for all examples.

中文翻译：

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具有不连续压力的两相 Stokes 问题的一种新的浸没式有限元方法

在本文中，我们针对两相不可压缩斯托克斯流开发了一种新的浸入式有限元法 (IFEM)。我们允许界面切割有限元。在非界面元素上，标准的 Crouzeix–Raviart 元素和 P 0 {P_{0}} 使用元素对。在界面元素上，为标量界面问题开发的基函数（Kwak 等人，破损分析 P 1个 {P_{1}} - 界面问题的不一致有限元法，暹罗 J. 数字。肛门。(2010)) 以速度和压力变量之间的耦合不同的方式进行修改。基函数有两种。第一类基在压力变量连续性的假设下满足拉普拉斯-杨条件。第二种，速度为气泡型，耦合不连续压力，仍满足拉普拉斯-杨条件。我们注意到，在第二种情况下，压力变量在每个界面元素上都有两个自由度。因此，我们的方法可以处理不连续压力的情况。提供了包括不连续压力变量情况在内的数值结果。我们看到了所有示例的最佳收敛顺序。