  In this paper, we present a numerical analysis of the hydrostatic Stokes
equations, which are linearization of the primitive equations describing the
geophysical flows of the ocean and the atmosphere. The hydrostatic Stokes
equations can be formulated as an abstract non-stationary saddle-point problem,
which also includes the non-stationary Stokes equations. We first consider the
finite element approximation for the abstract equations with a pair of spaces
under the discrete inf-sup condition. The aim of this paper is to establish
error estimates for the approximated solutions in various norms, in the
framework of analytic semigroup theory. Our main contribution is an error
estimate for the pressure with a natural singularity term $t^{-1}$, which is
induced by the analyticity of the semigroup. We also present applications of
the error estimates for the finite element approximations of the non-stationary
Stokes and the hydrostatic Stokes equations.
