  In this paper, we present a pressure-robust enriched Galerkin (EG) scheme for
solving the Stokes equations, which is an enhanced version of the EG scheme for
the Stokes problem proposed in [Son-Young Yi, Xiaozhe Hu, Sanghyun Lee, James
H. Adler, An enriched Galerkin method for the Stokes equations, Computers and
Mathematics with Applications, accepted, 2022]. The pressure-robustness is
achieved by employing a velocity reconstruction operator on the load vector on
the right-hand side of the discrete system. An a priori error analysis proves
that the velocity error is independent of the pressure and viscosity. We also
propose and analyze a perturbed version of our pressure-robust EG method that
allows for the elimination of the degrees of freedom corresponding to the
discontinuous component of the velocity vector via static condensation. The
resulting method can be viewed as a stabilized $H^1$-conforming
$\mathbb{P}_1$-$\mathbb{P}_0$ method. Further, we consider efficient block
preconditioners whose performances are independent of the viscosity. The
theoretical results are confirmed through various numerical experiments in two
and three dimensions.
