### 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.

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.

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Güemes 3450

S3000GLN Santa Fe, Argentina

Phone: 54-342-4511594 / 4511595 Int. 1006

Fax: 54-342-4511169

E-mail: amca(at)santafe-conicet.gov.ar

**Asociación Argentina de Mecánica Computacional**Güemes 3450

S3000GLN Santa Fe, Argentina

Phone: 54-342-4511594 / 4511595 Int. 1006

Fax: 54-342-4511169

E-mail: amca(at)santafe-conicet.gov.ar

**ISSN 2591-3522**