We develop a finite element scheme to approximate the dynamics of two and
three dimensional fluidic membranes in Navier–Stokes flow. Local inextensibility
of the membrane is ensured by solving a tangential Navier–Stokes equation, taking
surface viscosity effects of Boussinesq–Scriven type into account. In our approach
the bulk and surface degrees of freedom are discretized independently, which leads to
an unfitted finite element approximation of the underlying free boundary problem.
Bending elastic forces resulting from an elastic membrane energy are discretized using
an approximation introduced by Dziuk (2008). The obtained numerical scheme
can be shown to be stable and to have good mesh properties. Finally, the evolution
of membrane shapes is studied numerically in different flow situations in two and
three space dimensions. The numerical results demonstrate the robustness of the
method, and it is observed that the conservation properties are fulfilled to a high
precision.