  We present a novel monolithic divergence-conforming HDG scheme for a linear
fluid-structure interaction (FSI) problem with a thick structure. A
pressure-robust optimal energy-norm estimate is obtained for the semidiscrete
scheme. When combined with a Crank-Nicolson time discretization, our fully
discrete scheme is energy stable and produces an exactly divergence-free fluid
velocity approximation. The resulting linear system, which is symmetric and
indefinite, is solved using a preconditioned MinRes method with a robust block
algebraic multigrid (AMG) preconditioner.
