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On the error estimates of the vector penalty-projection methods: Second-order scheme


Authors: Philippe Angot and Rima Cheaytou
Journal: Math. Comp. 87 (2018), 2159-2187
MSC (2010): Primary 76D07, 35Q30, 65M15, 65M12
DOI: https://doi.org/10.1090/mcom/3309
Published electronically: December 27, 2017
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Abstract: In this paper, we study the vector penalty-projection method for incompressible unsteady Stokes equations with Dirichlet boundary conditions. The time derivative is approximated by the backward difference formula of second-order scheme (BDF2), namely Gear's scheme, whereas the approximation in space is performed by the finite volume scheme on a Marker And Cell (MAC) grid. After proving the stability of the method, we show that it yields second-error estimates in the time step for both velocity and pressure in the norm of $ l^{\infty }(\mathbf {L}^2(\Omega ))$ and $ l^2(L^2(\Omega ))$, respectively. Also, we show that the splitting error for both velocity and pressure is of order $ \mathcal {O}(\sqrt {\varepsilon \,\delta t})$ where $ \varepsilon $ is a penalty parameter chosen as small as desired and $ \delta t$ is the time step. Numerical results in agreement with the theoretical study are also provided.


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Additional Information

Philippe Angot
Affiliation: Aix-Marseille Université, Institut de Mathématiques de Marseille (I2M) - CNRS UMR7373, Centrale Marseille, 13453 Marseille cedex 13 - France
Email: philippe.angot@univ-amu.fr

Rima Cheaytou
Affiliation: Aix-Marseille Université, Institut de Mathématiques de Marseille (I2M) - CNRS UMR7373, Centrale Marseille, 13453 Marseille cedex 13 - France
Email: rima.cheaytou@gmail.com

DOI: https://doi.org/10.1090/mcom/3309
Received by editor(s): November 18, 2016
Received by editor(s) in revised form: April 12, 2017
Published electronically: December 27, 2017
Article copyright: © Copyright 2017 American Mathematical Society

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