  We present a parametric finite element approximation of two-phase flow with
insoluble surfactant. This free boundary problem is given by the Navier--Stokes
equations for the two-phase flow in the bulk, which are coupled to the
transport equation for the insoluble surfactant on the interface that separates
the two phases. We combine the evolving surface finite element method with an
approach previously introduced by the authors for two-phase Navier--Stokes
flow, which maintains good mesh properties. The derived finite element
approximation of two-phase flow with insoluble surfactant can be shown to be
stable. Several numerical simulations demonstrate the practicality of our
numerical method.
