Fluid Biomembranes: Modeling and Computation

We study two models for biomembranes. The first one is purely geometric since the equilibrium shapes are the minimizers of the Willmore energy under area and volume constraints. We present a novel method based on ideas from shape differential calculus. The second model incorporates the effect of the inside (bulk) viscous incompressible fluid and leads to more physical dynamics.

We use a parametric approach, which gives rise to fourth order highly nonlinear PDEs on surfaces and involves large domain deformations. We discretize these PDEs in space with an adaptive finite element method (AFEM), with either piecewise linear or quadratic polynomials, and a semi-implicit time stepping scheme. We employ the Taylor-Hood element for the Navier-Stokes equations together with iso-parametric elements, the latter being crucial for the correct approximation of curvature. We discuss several computational tools such as space-time adaptivity and mesh smoothing.

We also discuss a method to execute refinement, coarsening, and smoothing of meshes on manifolds with incomplete information about their geometry and yet preserve position and curvature accuracy. This is a new paradigm in adaptivity.

Ricardo Nochetto, University of Maryland, College Park

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