  We extend the recently introduced divergence-conforming immersed boundary
(DCIB) method [1] to fluid-structure interaction (FSI) problems involving
closed co-dimension one solids. We focus on capsules and vesicles, whose
discretization is particularly challenging due to the higher-order derivatives
that appear in their formulations. In two-dimensional settings, we employ cubic
B-splines with periodic knot vectors to obtain discretizations of closed curves
with C^2 inter-element continuity. In three-dimensional settings, we use
analysis-suitable bi-cubic T-splines to obtain discretizations of closed
surfaces with at least C^1 inter-element continuity. Large spurious changes of
the fluid volume inside closed co-dimension one solids is a well-known issue
for IB methods. The DCIB method results in volume changes orders of magnitude
lower than conventional IB methods. This is a byproduct of discretizing the
velocity-pressure pair with divergence-conforming B-splines, which lead to
negligible incompressibility errors at the Eulerian level. The higher
inter-element continuity of divergence-conforming B-splines is also crucial to
avoid the quadrature/interpolation errors of IB methods becoming the dominant
discretization error. Benchmark and application problems of vesicle and capsule
dynamics are solved, including mesh-independence studies and comparisons with
other numerical methods.
