  In this paper we study a system of advection-diffusion equations in a bulk
domain coupled to an advection-diffusion equation on an embedded surface. Such
systems of coupled partial differential equations arise in, for example, the
modeling of transport and diffusion of surfactants in two-phase flows. The
model considered here accounts for adsorption-desorption of the surfactants at
a sharp interface between two fluids and their transport and diffusion in both
fluid phases and along the interface. The paper gives a well-posedness analysis
for the system of bulk-surface equations and introduces a finite element method
for its numerical solution. The finite element method is unfitted, i.e., the
mesh is not aligned to the interface. The method is based on taking traces of a
standard finite element space both on the bulk domains and the embedded
surface. The numerical approach allows an implicit definition of the surface as
the zero level of a level-set function. Optimal order error estimates are
proved for the finite element method both in the bulk-surface energy norm and
the $L^2$-norm. The analysis is not restricted to linear finite elements and a
piecewise planar reconstruction of the surface, but also covers the
discretization with higher order elements and a higher order surface
reconstruction.
