A Doubly Adaptive Penalty Method for the Navier Stokes Equations

Xihui Xie, Kiera Kean, Shuxian Xu

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.

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