  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.
