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.
Accepté le :
DOI : 10.1051/m2an/2016028
Mots clés : Oseen equations, stabilized finite elements, fractional-step methods, pressure-correction methods, error estimates, high Reynolds number
@article{M2AN_2017__51_2_487_0, 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}, number = {2}, year = {2017}, doi = {10.1051/m2an/2016028}, mrnumber = {3626408}, zbl = {1398.76097}, language = {en}, url = {http://www.numdam.org/articles/10.1051/m2an/2016028/} }
TY - JOUR AU - Burman, Erik AU - Ern, Alexandre AU - Fernández, Miguel A. TI - Fractional-step methods and finite elements with symmetric stabilization for the transient Oseen problem JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2017 SP - 487 EP - 507 VL - 51 IS - 2 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2016028/ DO - 10.1051/m2an/2016028 LA - en ID - M2AN_2017__51_2_487_0 ER -
%0 Journal Article %A Burman, Erik %A Ern, Alexandre %A Fernández, Miguel A. %T Fractional-step methods and finite elements with symmetric stabilization for the transient Oseen problem %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2017 %P 487-507 %V 51 %N 2 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2016028/ %R 10.1051/m2an/2016028 %G en %F M2AN_2017__51_2_487_0
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://www.numdam.org/articles/10.1051/m2an/2016028/
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