  We develop, analyze and test adaptive penalty parameter methods. We prove
unconditional stability for velocity when adapting the penalty parameter,
$\epsilon,$ and stability of the velocity time derivative under a condition on
the change of the penalty parameter, $\epsilon(t_{n+1})-\epsilon(t_n)$. The
analysis and tests show that adapting $\epsilon(t_{n+1})$ in response to
$\nabla\cdot u(t_n)$ removes the problem of picking $\epsilon$ and yields good
approximations for the velocity. We provide error analysis and numerical tests
to support these results. We supplement the adaptive-$\epsilon$ method by also
adapting the time-step. The penalty parameter $\epsilon$ and time-step are
adapted independently. We further compare first, second and variable order
time-step algorithms. Accurate recovery of pressure remains an open problem.
