Finite Element Approximation for the Dynamics of Fluidic Two-Phase Biomembranes

TL;DR

This paper develops a finite element method to simulate the complex dynamics of fluidic two-phase biomembranes, capturing shape transitions driven by curvature and line energy effects.

Contribution

It introduces a stable semidiscrete finite element approximation for a coupled Cahn--Hilliard and Navier--Stokes model on evolving surfaces, enabling detailed simulations of membrane phenomena.

Findings

Successfully computed various two-phase membrane phenomena

Demonstrated stability and accuracy of the numerical method

Captured complex shape transitions driven by curvature and line energies

Abstract

Biomembranes and vesicles consisting of multiple phases can attain a multitude of shapes, undergoing complex shape transitions. We study a Cahn--Hilliard model on an evolving hypersurface coupled to Navier--Stokes equations on the surface and in the surrounding medium to model these phenomena. The evolution is driven by a curvature energy, modelling the elasticity of the membrane, and by a Cahn--Hilliard type energy, modelling line energy effects. A stable semidiscrete finite element approximation is introduced and, with the help of a fully discrete method, several phenomena occurring for two-phase membranes are computed.