  This paper introduces a new immersed boundary (IB) method for viscous
incompressible flow, based on a Fourier spectral method for the fluid solver
and on the nonuniform fast Fourier transform (NUFFT) algorithm for coupling the
fluid with the immersed boundary. The new Fourier spectral immersed boundary
(FSIB) method gives improved boundary resolution in comparison to the standard
IB method. The interpolated velocity field, in which the boundary moves, is
analytically divergence-free. The FSIB method is gridless and has the
meritorious properties of volume conservation, exact translation invariance,
conservation of momentum, and conservation of energy. We verify these
advantages of the FSIB method numerically both for the Stokes equations and for
the Navier-Stokes equations in both two and three space dimensions. The FSIB
method converges faster than the IB method. In particular, we observe
second-order convergence in various problems for the Navier-Stokes equations in
three dimensions. The FSIB method is also computationally efficient with
complexity of $O(N^3\log(N))$ per time step for $N^3$ Fourier modes in three
dimensions.
