  We present a new enriched Galerkin (EG) scheme for the Stokes equations based
on piecewise linear elements for the velocity unknowns and piecewise constant
elements for the pressure. The proposed EG method augments the conforming
piecewise linear space for velocity by adding an additional degree of freedom
which corresponds to one discontinuous linear basis function per element. Thus,
the total number of degrees of freedom is significantly reduced in comparison
with standard conforming, non-conforming, and discontinuous Galerkin schemes
for the Stokes equation. We show the well-posedness of the new EG approach and
prove that the scheme converges optimally. For the solution of the resulting
large-scale indefinite linear systems we propose robust block preconditioners,
yielding scalable results independent of the discretization and physical
parameters. Numerical results confirm the convergence rates of the
discretization and also the robustness of the linear solvers for a variety of
test problems.
