A stable numerical method for the dynamics of fluidic membranes

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 (Numer Math 111:55-80, 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.