Error analysis of the Penalty-Projection method for the time dependent Stokes equations

Abstract : We address in this paper a fractional-step scheme for the simulation of incompressible flows falling in the class of penalty-projection methods. The velocity prediction is similar to a penalty method prediction step, or, equivalently, differs from the incremental projection method one by the introduction of a penalty term built to enforce the divergence-free constraint. Then, a projection step based on a pressure Poisson equation is performed, to update the pressure and obtain an (approximately) divergence-free end-of-step velocity. An analysis in the energy norms for the model unsteady Stokes problem shows that this scheme enjoys the time convergence properties of both underlying methods: for low value of the penalty parameter r, the splitting error estimates of the so-called rotational projection scheme are recovered, i.e. convergence as $δt^2$ and $δt^{3/2}$ for the velocity and the pressure, respectively; for high values of the penalty parameter, we obtain the $δt/r$ behaviour for the velocity error known for the penalty scheme, together with a $1/r$ behaviour for the pressure error. Some numerical tests are presented, which substantiate this analysis.
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International Journal on Finite Volumes, Institut de Mathématiques de Marseille, AMU, 2009, 6 (1), pp.1-26
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Philippe Angot, Matthieu Jobelin, Jean-Claude Latché. Error analysis of the Penalty-Projection method for the time dependent Stokes equations. International Journal on Finite Volumes, Institut de Mathématiques de Marseille, AMU, 2009, 6 (1), pp.1-26. 〈hal-00482143〉

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