  Penalty methods relax the incompressibility condition and uncouple velocity
and pressure. Experience with them indicates that the velocity error is
sensitive to the choice of penalty parameter $\epsilon$. So far, there is no
effective \'a prior formula for $\epsilon$. Recently, Xie developed an adaptive
penalty scheme for the Stokes problem that picks the penalty parameter
$\epsilon$ self-adaptively element by element small where $\nabla \cdot u^h$ is
large. Her numerical tests gave accurate fluid predictions. The next natural
step, developed here, is to extend the algorithm with supporting analysis to
the non-linear, time-dependent incompressible Navier-Stokes equations. In this
report, we prove its unconditional stability, control of $\|\nabla \cdot
u^h\|$, and provide error estimates. We confirm the predicted convergence rates
with numerical tests.
