We consider the numerical approximation of a sharp-interface model for two-phase flow, which is given by the incompressible Navier-Stokes equations in the bulk domain together with the classical interface conditions on the interface. We propose structure-preserving finite element methods for the model, meaning in particular that volume preservation and energy decay are satisfied on the discrete level. For the evolving fluid interface, we employ parametric finite element approximations that introduce an implicit tangential velocity to improve the quality of the interface mesh. For the two-phase Navier-Stokes equations, we consider two different approaches: an unfitted and a fitted finite element method, respectively. In the unfitted approach, the constructed method is based on an Eulerian weak formulation, while in the fitted approach a novel arbitrary Lagrangian-Eulerian (ALE) weak formulation is introduced. Using suitable discretizations of these two formulations, we introduce two finite element methods and prove their structure-preserving properties. Numerical results are presented to show the accuracy and efficiency of the introduced methods.

翻译：我们考虑由不可压缩Navier-Stokes方程在体域内配合界面经典条件的尖锐界面两相流模型的数值逼近问题。针对该模型，我们提出了结构保持的有限元方法，特别地，该方法在离散层面满足体积守恒和能量衰减性质。对于运动的流体界面，我们采用参数化有限元逼近，通过引入隐式切向速度以改善界面网格质量。针对两相Navier-Stokes方程，我们分别考虑了非贴合与贴合两种有限元方法。非贴合方法基于欧拉弱形式构建，而贴合方法则引入了一种新颖的任意拉格朗日-欧拉（ALE）弱形式。通过采用合适的离散策略对这两种弱形式进行处理，我们提出了两种有限元方法，并证明了其结构保持特性。数值结果展示了所提方法的精度与计算效率。