A unified energy-stable finite element approximation for evolving fluidic biomembranes

We present a unified finite element method for the dynamics of fluidic biomembranes. The model is governed by the Navier–Stokes equations in the bulk coupled to the surface Navier–Stokes equations on the evolving biomembrane surface, with bending forces arising from the Willmore energy. By allowing the bulk mesh velocity to be independent of the fluid velocity and permitting a free tangential surface velocity, we are able to derive a unified weak formulation of the coupled bulk-surface Navier–Stokes system. To address the bending force, we consider an evolution equation for the curvature and propose a surface arbitrary Lagrangian–Eulerian (ALE) weak formulation. Discretization with either fitted or unfitted finite elements leads to well-posed fully discrete linear schemes that are unconditionally energy stable. We present a variety of numerical examples to demonstrate the favourable properties of the proposed methods.