Convergence of numerical approximations to a phase field bending elasticity model of membrane deformations

We study numerical approximations of a recently proposed phase field model for the vesicle membrane deformations governed by the variation of the elastic bending energy. Both the spatial discretization for the equilibrium problem with given volume and surface area constraints and the time discretization of a dynamic problem via gradient flow are considered. Convergence results of the numerical approximations are proved.

[1]  Qiang Du,et al.  Simulating the deformation of vesicle membranes under elastic bending energy in three dimensions , 2006, J. Comput. Phys..

[2]  Xiaoqiang Wang,et al.  PHASE FIELD MODELS AND SIMULATIONS OF VESICLE BIO-MEMBRANES , 2005 .

[3]  Udo Seifert,et al.  Configurations of fluid membranes and vesicles , 1997 .

[4]  L. R. Scott,et al.  The Mathematical Theory of Finite Element Methods , 1994 .

[5]  Qiang Du,et al.  A phase field formulation of the Willmore problem , 2005 .

[6]  Zhiming Chen,et al.  An error estimate for a finite-element scheme for a phase field model , 1994 .

[7]  Liyong Zhu,et al.  ANALYSIS OF A MIXED FINITE ELEMENT METHOD FOR A PHASE FIELD BENDING ELASTICITY MODEL OF VESICLE , 2006 .

[8]  Qiang Du,et al.  Analysis and Convergence of a Covolume Approximation of the Ginzburg--Landau Model of Superconductivity , 1998 .

[9]  W. Helfrich.  Elastic Properties of Lipid Bilayers: Theory and Possible Experiments , 1973, Zeitschrift fur Naturforschung. Teil C: Biochemie, Biophysik, Biologie, Virologie.

[10]  Jian Zhang,et al.  Adaptive Finite Element Method for a Phase Field Bending Elasticity Model of Vesicle Membrane Deformations , 2008, SIAM J. Sci. Comput..

[11]  Qiang Du,et al.  Modeling the spontaneous curvature effects in static cell membrane deformations by a phase field formulation , 2005 .

[12]  M. Chipot.  Finite Element Methods for Elliptic Problems , 2000 .

[13]  Numerical simulations of nonlinear phase transitions—I. The isotropic case , 1988 .

[14]  Chun Liu,et al.  Approximation of Liquid Crystal Flows , 2000, SIAM J. Numer. Anal..

[15]  Charles M. Elliott,et al.  Error estimates with smooth and nonsmooth data for a finite element method for the Cahn-Hilliard equation , 1992 .

[16]  Qiang Du,et al.  Retrieving Topological Information for Phase Field Models , 2005, SIAM J. Appl. Math..

[17]  Qiang Du,et al.  Semidiscrete Finite Element Approximations of a Linear Fluid-Structure Interaction Problem , 2004, SIAM J. Numer. Anal..

[18]  Charles M. Elliott,et al.  A second order splitting method for the Cahn-Hilliard equation , 1989 .

[19]  Gunduz Caginalp,et al.  A Numerical Analysis of an Anisotropic Phase Field Model , 1987 .

[20]  Q. Du,et al.  A phase field approach in the numerical study of the elastic bending energy for vesicle membranes , 2004 .

[21]  Andreas Prohl,et al.  Analysis of a fully discrete finite element method for the phase field model and approximation of its sharp interface limits , 2003, Math. Comput..

[22]  R. Nicolaides.  Direct discretization of planar div-curl problems , 1992 .

[23]  Ou-Yang Zhong-can,et al.  Geometric methods in the elastic theory of membranes in liquid crystal phases , 1999 .

[24]  R. Temam,et al.  Navier-Stokes equations: theory and numerical analysis: R. Teman North-Holland, Amsterdam and New York. 1977. 454 pp. US $45.00 , 1978 .

[25]  Qiang Du,et al.  Centroidal Voronoi Tessellations: Applications and Algorithms , 1999, SIAM Rev..