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This work deals with a posteriori error estimates for the Navier–Stokes equations. We propose a finite element discretization relying on the Galerkin method and we solve the discrete problem using an iterative method. Two sources of error appear, the discretization error and the linearization error. Balancing these two errors is very important to avoid performing an excessive number of iterations. Several numerical tests are provided to evaluate the efficiency of our indicators.
Bernardi, Christine 1 ; Dakroub, Jad 1, 2 ; Mansour, Gihane 2 ; Sayah, Toni 2
@article{M2AN_2016__50_4_1035_0,
author = {Bernardi, Christine and Dakroub, Jad and Mansour, Gihane and Sayah, Toni},
title = {A posteriori analysis of iterative algorithms for {Navier{\textendash}Stokes} problem},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1035--1055},
publisher = {EDP-Sciences},
volume = {50},
number = {4},
year = {2016},
doi = {10.1051/m2an/2015062},
zbl = {1457.65179},
mrnumber = {3521711},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/m2an/2015062/}
}
TY - JOUR AU - Bernardi, Christine AU - Dakroub, Jad AU - Mansour, Gihane AU - Sayah, Toni TI - A posteriori analysis of iterative algorithms for Navier–Stokes problem JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2016 SP - 1035 EP - 1055 VL - 50 IS - 4 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/m2an/2015062/ DO - 10.1051/m2an/2015062 LA - en ID - M2AN_2016__50_4_1035_0 ER -
%0 Journal Article %A Bernardi, Christine %A Dakroub, Jad %A Mansour, Gihane %A Sayah, Toni %T A posteriori analysis of iterative algorithms for Navier–Stokes problem %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2016 %P 1035-1055 %V 50 %N 4 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/m2an/2015062/ %R 10.1051/m2an/2015062 %G en %F M2AN_2016__50_4_1035_0
Bernardi, Christine; Dakroub, Jad; Mansour, Gihane; Sayah, Toni. A posteriori analysis of iterative algorithms for Navier–Stokes problem. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 50 (2016) no. 4, pp. 1035-1055. doi : 10.1051/m2an/2015062. http://geodesic.mathdoc.fr/articles/10.1051/m2an/2015062/
R.A. Adams, Sobolev Spaces. Acadamic Press, INC (1978). | Zbl
, A.H. Poza1 and F. Valentin, On a hierarchical error estimator combined with a stabilized method for the Navier–Stokes equations. Numer. Methods Partial Differ. Eq. 28 (2012) 782–806. | MR | DOI
and , Error estimates for adaptive finite element computations. SIAM J. Numer. Anal. 4 (1978) 736–754. | Zbl | MR | DOI
, and , On a higher order accurate fully discrete Galerkin approximation to the Navier–Stokes equations. Math. Comput. 39 (1982) 339–375. | Zbl | MR | DOI
and , Stream Function-Vorticity Driven Cavity Solution using p Finite Elements. Comput. Fluids 26 (1997) 453–468. | Zbl | DOI
, and , A finite element discretization of the three-dimensional Navier–Stokes equations with mixed boundary conditions. ESAIM: M2AN 43 (2009) 1185–1201. | Zbl | MR | mathdoc-id | DOI
, and , Finite dimensional approximation of nonlinear problems, Part I: Branches of nonsingular solutions. Numer. Math. 36 (1980) 1–25. | Zbl | MR | DOI
and , The 2D lid-driven cavity problem revisited. Comput. Fluids 35 (2006) 326–348. | Zbl | DOI
, Analytical and numerical studies of the structure of steady separated flows. J. Fluid Mech. 24 (1996) 113–151. | DOI
and , Computable error estimators for the approximation of nonlinear problems by linearized models. Comput. Methods Appl. Mech. Eng. 196 (2006) 210–224. | Zbl | MR | DOI
and , A posteriori estimation of the linearization error for strongly monotone nonlinear operators. Comput. Methods Appl. Mech. Eng. 205 (2007) 72–87. | Zbl | MR
, and , An a posteriori estimate for mixed finite element approximations of the Navier–Stokes equations. J. Korean Math. Soc. 48 (2011) 529–550. | Zbl | MR | DOI
, and , Guaranteed and robust a posteriori error estimates and balancing discretization and linearization errors for monotone nonlinear problems. Comput. Methods Appl. Mech. Eng. 200 (2011) 2782–2795. | Zbl | MR | DOI
and , Adaptive inexact Newton methods with a posteriori stopping criteria for nonlinear diffusion PDEs. SIAMJ. Sci. Comput. 35 (2013) A1761–A1791. | Zbl | MR | DOI
, Discussions on driven cavity flow. Int. J. Numer. Meth. Fluids 60 (2009) 747–774. | Zbl | MR | DOI
, and , Numerical solutions of 2-D steady incompressible driven cavity flow at high Reynolds numbers. Int. J. Numer. Methods Fluids 48 (2005) 747–774. | Zbl | DOI
V. Ervin, W. Layton and J. Maubach, A posteriori error estimators for a two-level finite element method for the Navier–Stokes equations. I.C.M.A. Tech. Report, University of Pittsburgh (1995) | Zbl | MR
V. Girault and P.-A. Raviart, Finite Element Methods for Navier–Stokes Equations. Springer-Verlag (1986). | Zbl | MR
, New development in FreeFem++. J. Numer. Math. 20 (2012) 251–266. | Zbl | MR | DOI
and , A posteriori error estimation in finite element analysis. Comput. Methods Appl. Mech. Eng. 159 (1998) 19–48. | Zbl | MR
, Residual a posteriori error estimates for two-level finite element methods for the Navier–Stokes equations. Appl. Numer. Math. 37 (2001) 501–518. | Zbl | MR | DOI
, Numerical Solution of the Navier–Stokes Equations for the Flow in a Two-Dimensional Cavity. J. Phys. Soc. Japan 16 (1961) 2307–2315. | Zbl | MR | DOI
O. Pironneau, Méthodes des éléments finis pour les fluides. Vol. 7 of Collection Recherches en Mathématiques Appliquées. Masson (1988). | Zbl
and , Consistency, stability, a priori and a posteriori errors for Petrov–Galerkin methods applied to nonlinear problems. Numer. Math. 69 (1994) 213–231. | Zbl | MR | DOI
and , Residual a posteriori error estimates for two-level finite element methods for the Navier–Stokes equations. Finite Elements in Analysis and Design 33 (1999) 247–262. | Zbl | MR | DOI
and , Driven cavity flows by efficient numerical techniques. J. Comput. Phys. 49 (1983) 310–333. | Zbl | DOI
R. Verfürth, A Posteriori Error Estimation Techniques For Finite Element Methods. Numer. Math. Sci. Comput. Oxford (2013). | Zbl | MR
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