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Int. J. Numer. Anal. Mod., 20 (2023), pp. 407-436.
Published online: 2023-03
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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.
}, issn = {2617-8710}, doi = {https://doi.org/10.4208/ijnam2023-1017}, url = {http://global-sci.org/intro/article_detail/ijnam/21540.html} }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.