  The immersed boundary (IB) method is an approach to fluid-structure
interaction that uses Lagrangian variables to describe the structure and
Eulerian variables to describe the fluid. Explicit time stepping schemes for
the IB method require solvers only for Eulerian equations, for which fast
Cartesian grid solution methods are available. Such methods are relatively
straightforward to develop and are widely used in practice but often require
very small time steps to maintain stability. Implicit-time IB methods permit
the stable use of large time steps, but efficient implementations of such
methods require significantly more complex solvers that effectively treat both
Lagrangian and Eulerian variables simultaneously. Several different approaches
to solving the coupled Lagrangian-Eulerian equations have been proposed, but a
complete understanding of this problem is still emerging. This paper presents a
geometric multigrid method for an implicit-time discretization of the IB
equations. This multigrid scheme uses a generalization of box relaxation that
is shown to handle problems in which the physical stiffness of the structure is
very large. Numerical examples are provided to illustrate the effectiveness and
efficiency of the algorithms described herein. These tests show that using
multigrid as a preconditioner for a Krylov method yields improvements in both
robustness and efficiency as compared to using multigrid as a solver. They also
demonstrate that with a time step 100--1000 times larger than that permitted by
an explicit IB method, the multigrid-preconditioned implicit IB method is
approximately 50--200 times more efficient than the explicit method.
