A Stabilized Maximum-Entropy Method for the Stokes Problem Coupled with a Phase-Field Model of Biomembranes

Abstract

Vesicles are closed biomembranes consisting of one or several different kinds of lipids. The stationary shapes of the vesicles are usually studied with the Canham-Helfrich bending energy model. We use a phase-field description of the membrane, governed by a fourth-order nonlinear partial differential equation with constraints. We tackle numerically this problem with the Local Maximum-Entropy (LME) approximants, since phase-field solutions benefit from the LME characteristics such as positivity, smoothness and variation diminishing property.
To analyze the dynamic properties of the vesicles, the fluid where they are immersed is commonly modeled as a Stokes flow because of the low Reynold’s number. The idea is to apply the same numerical scheme to compute both the phase-field bending energy and the bulk effect of the fluid field surrounding the membrane. It is well-known that the Stokes problem lacks pressure stability if velocity and pressure are described with the same interpolation space. This fact has led to two families of approaches, either using different and compatible spaces for the velocity and the pressure, or stabilizing equal interpolation methods. All these methods have been developed mainly in the context of finite elements. In this work we show stationary shapes of the vesicles computed with the phase-field approach and we present new results regarding the solution of Stokes benchmark problems using stabilized LME methods.