  Peskin's Immersed Boundary (IB) model and method are among one of the most
important modeling tools and numerical methods. The IB method has been known to
be first order accurate in the velocity. However, almost no rigorous
theoretical proof can be found in the literature for Stokes equations with a
prescribed velocity boundary condition. In this paper, it has been shown that
the pressure of the Stokes equation has a convergence order $O(\sqrt{h} |\log
h| )$ in the $L^2$ norm while the velocity has an $O(h |\log h| )$ convergence
order in the infinity norm in two-space dimensions. The proofs are based on
splitting the singular source terms, discrete Green functions on finite
lattices with homogeneous and Neumann boundary conditions, a new discovered
simplest $L^2$ discrete delta function, and the convergence proof of the IB
method for elliptic interface problems \cite{li:mathcom}. The conclusion in
this paper can apply to problems with different boundary conditions as long as
the problems are wellposed. The proof process also provides an efficient way to
decouple the system into three Helmholtz/Poisson equations without affecting
the order of convergence. A non-trivial numerical example is also provided to
confirm the theoretical analysis and the simple new discrete delta function.
