Fractional-step methods and finite elements with symmetric stabilization for the transient Oseen problem
Burman, Erik1 ; Ern, Alexandre2 ; Fernández, Miguel A.3, 4
1 Department of Mathematics, University College London, Gower Street, London, WC1E 6BT, UK.
2 Université Paris-Est, CERMICS (ENPC), 77455 Marne la Vallée cedex 2, France.
3 Inria, 75012 Paris, France.
4 Sorbonnes Universités, UPMC, Laboratoire Jacques-Louis Lions, 75005 Paris, France.
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 2, pp. 487-507.

This paper deals with the spatial and time discretization of the transient Oseen equations. Finite elements with symmetric stabilization in space are combined with several time-stepping schemes (monolithic and fractional-step). Quasi-optimal (in space) and optimal (in time) error estimates are established for smooth solutions in all flow regimes. We first analyze monolithic time discretizations using the Backward Differentation Formulas of order 1 and 2 (BDF1 and BDF2). We derive a new estimate on the time-average of the pressure error featuring the same robustness with respect to the Reynolds number as the velocity estimate. Then, we analyze fractional-step pressure-projection methods using BDF1. The stabilization of velocities and pressures can be treated either implicitly or explicitly. Numerical results illustrate the main theoretical findings.

Reçu le :
Accepté le :
DOI : 10.1051/m2an/2016028
Classification : 65M12, 65M60, 76D07, 76M10
Mots-clés : Oseen equations, stabilized finite elements, fractional-step methods, pressure-correction methods, error estimates, high Reynolds number
Burman, Erik 1 ; Ern, Alexandre 2 ; Fernández, Miguel A. 3, 4

1 Department of Mathematics, University College London, Gower Street, London, WC1E 6BT, UK.
2 Université Paris-Est, CERMICS (ENPC), 77455 Marne la Vallée cedex 2, France.
3 Inria, 75012 Paris, France.
4 Sorbonnes Universités, UPMC, Laboratoire Jacques-Louis Lions, 75005 Paris, France. 
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     author = {Burman, Erik and Ern, Alexandre and Fern\'andez, Miguel A.},
     title = {Fractional-step methods and finite elements with symmetric stabilization for the transient {Oseen} problem},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {487--507},
     publisher = {EDP-Sciences},
     volume = {51},
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Burman, Erik; Ern, Alexandre; Fernández, Miguel A. Fractional-step methods and finite elements with symmetric stabilization for the transient Oseen problem. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 2, pp. 487-507. doi : 10.1051/m2an/2016028. http://archive.numdam.org/articles/10.1051/m2an/2016028/

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