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John W. Barrett; Harald Garcke; Robert Nürnberg
Gradient flow dynamics of two-phase biomembranes: Sharp interface variational formulation and finite element approximation
SMAI-Journal of computational mathematics, 4 (2018), p. 151-195, doi: 10.5802/smai-jcm.32
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Class. Math.: 35R01, 49Q10, 65M12, 65M60, 82B26, 92C10
Keywords: parametric finite elements, Helfrich energy, spontaneous curvature, multi-phase membrane, line energy, $C^0$– and $C^1$–matching conditions

Abstract

A finite element method for the evolution of a two-phase membrane in a sharp interface formulation is introduced. The evolution equations are given as an $L^2$–gradient flow of an energy involving an elastic bending energy and a line energy. In the two phases Helfrich-type evolution equations are prescribed, and on the interface, an evolving curve on an evolving surface, highly nonlinear boundary conditions have to hold. Here we consider both $C^0$– and $C^1$–matching conditions for the surface at the interface. A new weak formulation is introduced, allowing for a stable semidiscrete parametric finite element approximation of the governing equations. In addition, we show existence and uniqueness for a fully discrete version of the scheme. Numerical simulations demonstrate that the approach can deal with a multitude of geometries. In particular, the paper shows the first computations based on a sharp interface description, which are not restricted to the axisymmetric case.

Bibliography

[1] H. Abels, H. Garcke & L. Müller, “Local well-posedness for volume-preserving mean curvature and Willmore flows with line tension”, Math. Nachr. 289 (2016) no. 2–3, p. 136-174 Article
[2] J. W. Barrett, H. Garcke & R. Nürnberg, “A parametric finite element method for fourth order geometric evolution equations”, J. Comput. Phys. 222 (2007) no. 1, p. 441-462 Article
[3] J. W. Barrett, H. Garcke & R. Nürnberg, “On the parametric finite element approximation of evolving hypersurfaces in ${\mathbb{R}}^3$”, J. Comput. Phys. 227 (2008) no. 9, p. 4281-4307 Article
[4] J. W. Barrett, H. Garcke & R. Nürnberg, “Parametric Approximation of Surface Clusters driven by Isotropic and Anisotropic Surface Energies”, Interfaces Free Bound. 12 (2010) no. 2, p. 187-234 Article
[5] J. W. Barrett, H. Garcke & R. Nürnberg, “The Approximation of Planar Curve Evolutions by Stable Fully Implicit Finite Element Schemes that Equidistribute”, Numer. Methods Partial Differential Equations 27 (2011) no. 1, p. 1-30 Article
[6] J. W. Barrett, H. Garcke & R. Nürnberg, “Elastic flow with junctions: Variational approximation and applications to nonlinear splines”, Math. Models Methods Appl. Sci. 22 (2012) no. 11 Article
[7] J. W. Barrett, H. Garcke & R. Nürnberg, “On the Stable Numerical Approximation of Two-Phase Flow with Insoluble Surfactant”, M2AN Math. Model. Numer. Anal. 49 (2015) no. 2, p. 421-458 Article
[8] J. W. Barrett, H. Garcke & R. Nürnberg, “Computational Parametric Willmore Flow with Spontaneous Curvature and Area Difference Elasticity effects”, SIAM J. Numer. Anal. 54 (2016) no. 3, p. 1732-1762 Article
[9] J. W. Barrett, H. Garcke & R. Nürnberg, “Finite Element Approximation for the Dynamics of Fluidic Two-Phase Biomembranes”, M2AN Math. Model. Numer. Anal. 51 (2017) no. 6, p. 2319-2366 Article
[10] J. W. Barrett, H. Garcke & R. Nürnberg, “Stable variational approximations of boundary value problems for Willmore flow with Gaussian curvature”, IMA J. Numer. Anal. 37 (2017) no. 4, p. 1657-1709 Article
[11] T. Baumgart, S. Das, W. W. Webb & J.T. Jenkins, “Membrane Elasticity in Giant Vesicles with Fluid Phase Coexistence”, Biophys. J. 89 (2005) no. 2, p. 1067-1080 Article
[12] T. Baumgart, Hess; S. T. & W. W. Webb, “Imaging coexisting fluid domains in biomembrane models coupling curvature and line tension”, Nature 425 (2003) no. 6960, p. 821-824 Article
[13] R. Choksi, M. Morandotti & M. Veneroni, “Global minimizers for axisymmetric multiphase membranes”, ESAIM Control Optim. Calc. Var. 19 (2013) no. 4, p. 1014-1029 Article
[14] G. Cox & J. Lowengrub, “The effect of spontaneous curvature on a two-phase vesicle”, Nonlinearity 28 (2015) no. 3, p. 773-793 Article
[15] S.L. Das, J.T. Jenkins & T. Baumgart, “Neck geometry and shape transitions in vesicles with co-existing fluid phases: Role of Gaussian curvature stiffness vs. spontaneous curvature”, Europhys. Lett. 86 (2009) no. 4
[16] T. A. Davis, “Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method”, ACM Trans. Math. Software 30 (2004) no. 2, p. 196-199 Article
[17] K. Deckelnick, G. Dziuk & C. M. Elliott, “Computation of geometric partial differential equations and mean curvature flow”, Acta Numer. 14 (2005), p. 139-232 Article
[18] K. Deckelnick, H.-C. Grunau & M. Röger, “Minimising a relaxed Willmore functional for graphs subject to boundary conditions”, Interfaces Free Bound. 19 (2017) no. 1, p. 109-140
[19] G. Dziuk, “An algorithm for evolutionary surfaces”, Numer. Math. 58 (1991) no. 6, p. 603-611 Article
[20] G. Dziuk, “Computational parametric Willmore flow”, Numer. Math. 111 (2008) no. 1, p. 55-80 Article
[21] G. Dziuk & C. M. Elliott, “Finite element methods for surface PDEs”, Acta Numer. 22 (2013), p. 289-396
[22] C. M. Elliott & B. Stinner, “Modeling and computation of two phase geometric biomembranes using surface finite elements”, J. Comput. Phys. 229 (2010) no. 18, p. 6585-6612 Article
[23] C. M. Elliott & B. Stinner, “A surface phase field model for two-phase biological membranes”, SIAM J. Appl. Math. 70 (2010) no. 8, p. 2904-2928 Article
[24] C.M. Elliott & B. Stinner, “Computation of two-phase biomembranes with phase dependent material parameters using surface finite elements”, Commun. Comput. Phys. 13 (2013) no. 2, p. 325-360
[25] M. Helmers, “Snapping elastic curves as a one-dimensional analogue of two-component lipid bilayers”, Math. Models Methods Appl. Sci. 21 (2011) no. 5, p. 1027-1042
[26] M. Helmers, “Kinks in two-phase lipid bilayer membranes”, Calc. Var. Partial Differential Equations 48 (2013) no. 1-2, p. 211-242 Article
[27] M. Helmers, “Convergence of an approximation for rotationally symmetric two-phase lipid bilayer membranes”, Q. J. Math. 66 (2015) no. 1, p. 143-170 Article
[28] F. Jülicher & R. Lipowsky, “Domain-induced budding of vesicles”, Phys. Rev. Lett. 70 (1993) no. 19, p. 2964-2967
[29] F. Jülicher & R. Lipowsky, “Shape transformations of vesicles with intramembrane domains”, Phys. Rev. E 53 (1996) no. 3, p. 2670-2683 Article
[30] J. S. Lowengrub, A. Rätz & A. Voigt, “Phase-field modeling of the dynamics of multicomponent vesicles: Spinodal decomposition, coarsening, budding, and fission”, Phys. Rev. E 79 (2009) no. 3
[31] M. Mercker & A. Marciniak-Czochra, “Bud-Neck Scaffolding as a Possible Driving Force in ESCRT-Induced Membrane Budding”, Biophys. J. 108 (2015) no. 4, p. 833-843 Article
[32] M. Mercker, A. Marciniak-Czochra, T. Richter & D. Hartmann, “Modeling and computing of deformation dynamics of inhomogeneous biological surfaces”, SIAM J. Appl. Math. 73 (2013) no. 5, p. 1768-1792 Article
[33] J. C. C. Nitsche, “Boundary value problems for variational integrals involving surface curvatures”, Quart. Appl. Math. 51 (1993) no. 2, p. 363-387
[34] A. Schmidt & K. G. Siebert, Design of Adaptive Finite Element Software: The Finite Element Toolbox ALBERTA, Lecture Notes in Computational Science and Engineering 42, Springer-Verlag, Berlin, 2005
[35] S. Schmidt & V. Schulz, “Shape derivatives for general objective functions and the incompressible Navier–Stokes equations”, Control Cybernet. 39 (2010) no. 3, p. 677-713
[36] M. E. Taylor, Partial differential equations I. Basic theory, Applied Mathematical Sciences 115, Springer, 2011
[37] F. Tröltzsch, Optimal Control of Partial Differential Equations: Theory, Methods and Applications, Graduate Studies in Mathematics 112, American Mathematical Society, Providence, RI, 2010
[38] Z.-C. Tu, “Challenges in theoretical investigations of configurations of lipid membranes”, Chin. Phys. B 22 (2013) no. 2 Article
[39] Z. C. Tu & Z. C. Ou-Yang, “A geometric theory on the elasticity of bio-membranes”, J. Phys. A 37 (2004) no. 47, p. 11407-11429 Article
[40] X. Wang & Q. Du, “Modelling and simulations of multi-component lipid membranes and open membranes via diffuse interface approaches”, J. Math. Biol. 56 (2008) no. 3, p. 347-371 Article
[41] C. Wutz, Variationsprobleme für elastische Biomembranen unter Berücksichtigung von Linienenergien, Diploma thesis, University Regensburg, 2010