We propose a new unfitted finite element method for simulation of two-phase flows in presence of insoluble surfactant. The key features of the method are 1) discrete conservation of surfactant mass; 2) the possibility of having meshes that do not conform to the evolving interface separating the immiscible fluids; 3) accurate approximation of quantities with weak or strong discontinuities across evolving geometries such as the velocity field and the pressure. The new discretization of the incompressible Navier--Stokes equations coupled to the convection-diffusion equation modeling the surfactant transport on evolving surfaces is based on a space-time cut finite element formulation with quadrature in time and a stabilization term in the weak formulation that provides function extension. The proposed strategy utilize the same computational mesh for the discretization of the surface Partial Differential Equation (PDE) and the bulk PDEs and can be combined with different techniques for representing and evolving the interface, here the level set method is used. Numerical simulations in both two and three space dimensions are presented including simulations showing the role of surfactant in the interaction between two drops.

翻译：我们提出一种新的不适宜有限元素方法,用于模拟在有不可溶性表面活性剂的情况下的两阶段流动。该方法的主要特征是:(1) 离散地保护表面活性质量;(2) 有可能使与不断演变的界面不兼容的不易流体;(3) 精确接近数量,在速度场和压力等不断变化的地理配方体中,出现微弱或强烈的不连续现象;在不断演变的表面上,将不可压缩的导航-斯托克方程式与对流-扩散方程式进行新的分解,同时对冲动方程式进行模型的分解。在不断演变的表面表面和压力上,采用水平设定方法,以时空时间缩短的有限元素配方和提供功能延伸的弱方形中的稳定性术语为基础。拟议战略使用相同的计算网块,对地表部分差异化(PDE)和大片PDE的离散作用进行计算,并与代表界面和演进的不同技术相结合。这里使用的是水平定法方法。在两个和三个空间维的数值模拟中,都展示了在两个空间层面中显示表面活性相互作用的作用。