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A connection between coupled and penalty projection timestepping schemes with FE spatial discretization for the Navier–Stokes equations

Alexander Linke, Michael Neilan, Leo G. Rebholz and Nicholas E. Wilson

Abstract

We prove that for several inf-sup stable mixed finite elements, the solution of the Chorin/Temam projection methods for Navier–Stokes equations equipped with grad–div stabilization with parameter γ converge to the associated coupled method solution with rate γ−1 as γ → ∞. We prove this result for both backward Euler schemes and BDF2 schemes. Furthermore, we simplify classical numerical analysis of projection methods, allowing us to remove some unnecessary assumptions, such as convexity of the domain. Several numerical experiments are given which verify the convergence rate, and show that projection methods with large grad–div stabilization parameters can dramatically improve accuracy.

MSC 2010: 65M60; 76D05

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Received: 2016-4-2
Revised: 2016-6-15
Accepted: 2016-7-3
Published Online: 2018-2-19
Published in Print: 2017-12-20

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