A Doubly Adaptive Penalty Method for the Navier Stokes Equations

We develop, analyze and test adaptive penalty parameter methods. We prove unconditional stability for velocity when adapting the penalty parameter, $ϵ,$ and stability of the velocity time derivative under a condition on the change of the penalty parameter, $ϵ(t_{n+1})-ϵ(t_n)$. The analysis and tests show that adapting $ϵ(t_{n+1})$ in response to $\mathrm{\nabla }\cdot u(t_n)$ removes the problem of picking $ϵ$ and yields good approximations for the velocity. We provide error analysis and numerical tests to support these results. We supplement the adaptive-$ϵ$ method by also adapting the time-step. The penalty parameter $ϵ$ 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.