  We present a numerical stability analysis of the immersed boundary(IB) method
for a special case which is constructed so that Fourier analysis is applicable.
We examine the stability of the immersed boundary method with the discrete
Fourier transforms defined differently on the fluid grid and the boundary grid.
This approach gives accurate theoretical results about the stability boundary
since it takes the effects of the spreading kernel of the immersed boundary
method on the numerical stability into account. In this paper, the spreading
kernel is the standard 4-point IB delta function. A three-dimensional
incompressible viscous flow and a no-slip planar boundary are considered. The
case of a planar elastic membrane is also analyzed using the same analysis
framework and it serves as an example of many possible generalizations of our
theory. We present some numerical results and show that the observed stability
behaviors are consistent with what are predicted by our theory.
