Authors: Shravan K. Veerapaneni, Denis Gueyerer, George Biros, Denis Zorin

Title: A numerical method for simulating the dynamics of 3D
axisymmetric vesicles suspended in viscous flows

 We extend "A boundary integral method for simulating the dynamics of inextensible vesicles suspended
 in a viscous fluid in 2D", Veerapaneni et al. Journal of Computational Physics, 228(7), 2009 to the
 case of three dimensional axisymmetric vesicles of spherical or toroidal topology immersed in viscous
 flows. Although the main components of the algorithm are similar in spirit to the 2D case.spectral
 approximation in space, semi-implicit time-stepping scheme.the main differences are that the bending
 and viscous force require new analysis, the linearization for the semi-implicit schemes must be rederived,
 a fully implicit scheme must be used for the toroidal topology to eliminate a CFL-type restriction, and
 a novel numerical scheme for the evaluation of the 3D Stokes single-layer potential on an axisymmetric
 surface is necessary to speed up the calculations. By introducing these novel components, we obtain a
 time-scheme that experimentally is unconditionally stable, has low cost per time step, and is third-order
 accurate in time. We present numerical results to analyze the cost and convergence rates of the scheme.
 To verify the solver, we compare it to a constrained variational approach to compute equilibrium shapes
 that does not involve interactions with a viscous fluid. To illustrate the applicability of method, we
 consider a few vesicle-flow interaction problems: the sedimentation of a vesicle, interactions of one and
 three vesicles with a background Poiseuille flow.