A unified energy-stable finite element approximation for evolving fluidic biomembranes
This work provides a robust numerical framework for simulating biomembrane dynamics, which is important for biophysics and computational biology.
The paper develops a unified finite element method for simulating evolving fluidic biomembranes, coupling bulk and surface Navier-Stokes equations with bending forces from Willmore energy. The method is unconditionally energy stable and demonstrated through numerical examples.
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.