The shapes of vesicles formed by lipid bilayers with phase separation are governed by a bending energy with phase dependent material parameters together with a line energy associated with the phase interfaces. We present a numerical method to approximate solutions to the Euler-Lagrange equations featuring triangulated surfaces, isoparametric quadratic surface finite elements and the phase field approach for the phase separation. Furthermore, the method involves an iterative solution scheme that is based on a relaxation dynamics coupling a geometric evolution equation for the membrane surface with a surface Allen-Cahn equation. Remeshing and grid adaptivity are discussed, and in various simulations the influence of several physical parameters is investigated.

We derive a biomembrane model consisting of a fluid enclosed by a lipid membrane. Themembrane is characterized by its Canham-Helfrich energy (Willmore energy with areaconstraint) and acts as a boundary force on the Navier-Stokes system modeling anincompressible fluid. We give a concise description of the model and of the associatednumerical scheme. We provide numerical simulations with emphasis on the comparisonsbetween different types of flow: the geometric model which does not take into account thebulk fluid and the biomembrane model for two different regimes of parameters.

We investigate the evolution of an almost flat membrane driven by competition of the homogeneous, Frank, and bending energies as wellas the coupling of the local order of the constituent molecules of the membrane to its curvature.We propose an alternative to the model in [J.B. Fournier and P. Galatoa, J. Phys. II7 (1997) 1509–1520; N. Uchida, Phys. Rev. E66 (2002) 040902] which replacesa Ginzburg-Landau penalization for the length of theorder parameter by a rigid constraint.We introduce a fully discrete scheme, consisting of piecewise linearfinite elements, show that it is unconditionally stable for a large range of the elastic moduli in the model, and prove its convergence(up to subsequences) thereby proving the existence of a weak solutionto the continuous model. Numerical simulations are included that examine typical model situations, confirm our theory, and explore numerical predictions beyond that theory.