  The Navier-Stokes-$\alpha$ equations belong to the family of LES (Large Eddy
Simulation) models whose fundamental idea is to capture the influence of the
small scales on the large ones without computing all the whole range present in
the flow. The constant $\alpha$ is a regime flow parameter that has the
dimension of the smallest scale being resolvable by the model. Hence, when
$\alpha=0$, one recovers the classical Navier-Stokes equations for a flow of
viscous, incompressible, Newtonian fluids. Furthermore, the
Navier-Stokes-$\alpha$ equations can also be interpreted as a regularization of
the Navier-Stokes equations, where $\alpha$ stands for the regularization
parameter.
  In this paper we first present the Navier-Stokes-$\alpha$ equations on
bounded domains with no-slip boundary conditions by means of the Leray
regularization using the Helmholtz operator. Then we study the problem of
relating the behavior of the Galerkin approximations for the
Navier-Stokes-$\alpha$ equations to that of the solutions of the Navier-Stokes
equations on bounded domains with no-slip boundary conditions. The Galerkin
method is undertaken by using the eigenfunctions associated with the Stokes
operator. We will derive local- and global-in-time error estimates measured in
terms of the regime parameter $\alpha$ and the eigenvalues. In particular, in
order to obtain global-in-time error estimates, we will work with the concept
of stability for solutions of the Navier-Stokes equations in terms of the $L^2$
norm.
