Error estimates with optimal convergence orders are proved for a stabilized Lagrange−Galerkin scheme for the Navier−Stokes equations. The scheme is a combination of Lagrange−Galerkin method and Brezzi−Pitkäranta’s stabilization method. It maintains the advantages of both methods; (i) It is robust for convection-dominated problems and the system of linear equations to be solved is symmetric. (ii) Since the P1 finite element is employed for both velocity and pressure, the number of degrees of freedom is much smaller than that of other typical elements for the equations, e.g., P2/P1. Therefore, the scheme is efficient especially for three-dimensional problems. The theoretical convergence orders are recognized numerically by two- and three-dimensional computations.
DOI : 10.1051/m2an/2015047
Mots clés : Error estimates, the finite element method, the Lagrange−Galerkin method, pressure-stabilization, the Navier−Stokes equations
@article{M2AN_2016__50_2_361_0,
author = {Notsu, Hirofumi and Tabata, Masahisa},
title = {Error estimates of a stabilized {Lagrange\ensuremath{-}Galerkin} scheme for the {Navier\ensuremath{-}Stokes} equations},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {361--380},
publisher = {EDP-Sciences},
volume = {50},
number = {2},
year = {2016},
doi = {10.1051/m2an/2015047},
mrnumber = {3482547},
zbl = {1381.76192},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2015047/}
}
TY - JOUR AU - Notsu, Hirofumi AU - Tabata, Masahisa TI - Error estimates of a stabilized Lagrange−Galerkin scheme for the Navier−Stokes equations JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2016 SP - 361 EP - 380 VL - 50 IS - 2 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2015047/ DO - 10.1051/m2an/2015047 LA - en ID - M2AN_2016__50_2_361_0 ER -
%0 Journal Article %A Notsu, Hirofumi %A Tabata, Masahisa %T Error estimates of a stabilized Lagrange−Galerkin scheme for the Navier−Stokes equations %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2016 %P 361-380 %V 50 %N 2 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2015047/ %R 10.1051/m2an/2015047 %G en %F M2AN_2016__50_2_361_0
Notsu, Hirofumi; Tabata, Masahisa. Error estimates of a stabilized Lagrange−Galerkin scheme for the Navier−Stokes equations. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 50 (2016) no. 2, pp. 361-380. doi : 10.1051/m2an/2015047. http://www.numdam.org/articles/10.1051/m2an/2015047/
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