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.