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：本文考虑由不可压缩Navier-Stokes方程（在体域内）及经典界面条件（在界面上）给出的尖锐界面两相流模型的数值逼近。我们为该模型提出保持结构特性的有限元方法，其核心在于离散层面满足体积守恒与能量衰减性质。针对演化的流体界面，采用引入隐式切向速度以提高界面网格质量的参数化有限元逼近。对于两相Navier-Stokes方程，分别考虑非拟合与拟合两类有限元方法：非拟合方法基于欧拉弱形式构建，而拟合方法则引入新型任意拉格朗日-欧拉（ALE）弱形式。通过对这两种弱形式进行适当离散化，提出两种有限元方法并证明其结构保持特性。数值结果展示了所提方法的精度与计算效率。