  In this paper we present a finite element analysis for a Dirichlet boundary
control problem governed by the Stokes equation. The Dirichlet control is
considered in a convex closed subset of the energy space
$\mathbf{H}^1(\Omega).$ Most of the previous works on the Stokes Dirichlet
boundary control problem deals with either tangential control or the case where
the flux of the control is zero. This choice of the control is very particular
and their choice of the formulation leads to the control with limited
regularity. To overcome this difficulty, we introduce the Stokes problem with
outflow condition and the control acts on the Dirichlet boundary only hence our
control is more general and it has both the tangential and normal components.
We prove well-posedness and discuss on the regularity of the control problem.
The first-order optimality condition for the control leads to a Signorini
problem. We develop a two-level finite element discretization by using
$\mathbf{P}_1$ elements(on the fine mesh) for the velocity and the control
variable and $P_0$ elements (on the coarse mesh) for the pressure variable. The
standard energy error analysis gives $\frac{1}{2}+\frac{\delta}{2}$ order of
convergence when the control is in $\mathbf{H}^{\frac{3}{2}+\delta}(\Omega)$
space. Here we have improved it to $\frac{1}{2}+\delta,$ which is optimal.
Also, when the control lies in less regular space we derived optimal order of
convergence up to the regularity. The theoretical results are corroborated by a
variety of numerical tests.
