A Doubly Adaptive Penalty Method for the Navier Stokes Equations

01/11/2022

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by   Xihui Xie, et al.

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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 ∇· 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.