  We develop an immersed boundary (IB) method for modeling flows around fixed
or moving rigid bodies that is suitable for a broad range of Reynolds numbers,
including steady Stokes flow. The spatio-temporal discretization of the fluid
equations is based on a standard staggered-grid approach. Fluid-body
interaction is handled using Peskin's IB method; however, unlike existing IB
approaches to such problems, we do not rely on penalty or fractional-step
formulations. Instead, we use an unsplit scheme that ensures the no-slip
constraint is enforced exactly in terms of the Lagrangian velocity field
evaluated at the IB markers. Fractional-step approaches, by contrast, can
impose such constraints only approximately. Imposing these constraints exactly
requires the solution of a large linear system that includes the fluid velocity
and pressure as well as Lagrange multiplier forces that impose the motion of
the body. To solve this system efficiently, we develop a preconditioner for the
constrained IB formulation that is based on an analytical approximation to the
Schur complement. This approach is enabled by the near translational and
rotational invariance of Peskin's IB method. We demonstrate that only a few
cycles of a geometric multigrid method for the fluid equations are required in
each application of the preconditioner, and we demonstrate robust convergence
of the overall Krylov solver despite the approximations made in the
preconditioner. We apply the method to a number of test problems at zero and
finite Reynolds numbers, and we demonstrate first-order convergence of the
method to several analytical solutions and benchmark computations.
