  Penalizing incompressibility in the Stokes problem leads, under mild
assumptions, to matrices with condition numbers $\kappa =\mathcal{O}
(\varepsilon ^{-1}h^{-2})$, $\varepsilon =$ penalty parameter $<<1$, and $ h= $
mesh width $<1$. Although $\kappa =\mathcal{O}(\varepsilon ^{-1}h^{-2}) $ is
large, practical tests seldom report difficulty in solving these systems. In
the SPD case, using the conjugate gradient method, this is usually explained by
spectral gaps occurring in the penalized coefficient matrix. Herein we point
out a second contributing factor. Since the solution is approximately
incompressible, solution components in the eigenspaces associated with the
penalty terms can be small. As a result, the effective condition number can be
much smaller than the standard condition number.
