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：本文研究尖界面两相流模型的数值逼近问题，该模型由体域内的不可压缩Navier-Stokes方程及界面上的经典界面条件共同描述。我们针对该模型提出保结构有限元方法，其核心在于离散层次上满足体积守恒与能量耗散特性。针对演化的流体界面，我们采用参数化有限元逼近方法，通过引入隐式切向速度提升界面网格质量。对于两相Navier-Stokes方程，本文分别考虑两种不同途径：非拟合方法与拟合有限元法。在非拟合方法中，所构建方法基于欧拉弱形式；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恰当离散，我们提出两种有限元方法并证明其保结构性质。数值实验结果展示了所提方法的精确性与计算效率。