no. 4
A posteriori analysis of iterative algorithms for Navier–Stokes problem
Bernardi, Christine1  ; Dakroub, Jad1, 2  ; Mansour, Gihane2  ; Sayah, Toni2 
1 Laboratoire Jacques-Louis Lions - C.N.R.S. et Université Pierre et Marie Curie, 4 place Jussieu, 75252 Paris cedex 05, France.
2 Unité de recherche EGFEM, Faculté des sciences, Université Saint-Joseph, Lebanon, Beirut, Liban.
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 50 (2016) no. 4, pp. 1035-1055

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.

Reçu le :
DOI : 10.1051/m2an/2015062
Classification : 65N30, 65N15, 65J15, 76D05
Keywords: A posteriori error estimation, Navier–Stokes problem, iterative method

Bernardi, Christine 1 ; Dakroub, Jad 1, 2 ; Mansour, Gihane 2 ; Sayah, Toni 2

1 Laboratoire Jacques-Louis Lions - C.N.R.S. et Université Pierre et Marie Curie, 4 place Jussieu, 75252 Paris cedex 05, France.
2 Unité de recherche EGFEM, Faculté des sciences, Université Saint-Joseph, Lebanon, Beirut, Liban. 
@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},
     year = {2016},
     publisher = {EDP Sciences},
     volume = {50},
     number = {4},
     doi = {10.1051/m2an/2015062},
     zbl = {1457.65179},
     mrnumber = {3521711},
     language = {en},
     url = {https://www.numdam.org/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  - https://www.numdam.org/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 https://www.numdam.org/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

R.A. Adams, Sobolev Spaces. Acadamic Press, INC (1978). | Zbl

R. Araya, 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

I. Babuška and W.C. Rheinboldt, Error estimates for adaptive finite element computations. SIAM J. Numer. Anal. 4 (1978) 736–754. | Zbl | MR | DOI

A. Baker, V.A. Dougalis and O.A. Karakashian, On a higher order accurate fully discrete Galerkin approximation to the Navier–Stokes equations. Math. Comput. 39 (1982) 339–375. | Zbl | MR | DOI

E. Barragy and G.F. Carey, Stream Function-Vorticity Driven Cavity Solution using p Finite Elements. Comput. Fluids 26 (1997) 453–468. | Zbl | DOI

C. Bernardi, F. Hecht and R. Verfürth, A finite element discretization of the three-dimensional Navier–Stokes equations with mixed boundary conditions. ESAIM: M2AN 43 (2009) 1185–1201. | Zbl | MR | Numdam | DOI

F. Brezzi, J. Rappaz and P.-A. Raviart, Finite dimensional approximation of nonlinear problems, Part I: Branches of nonsingular solutions. Numer. Math. 36 (1980) 1–25. | Zbl | MR | DOI

C.H. Bruneau and M. Saad, The 2D lid-driven cavity problem revisited. Comput. Fluids 35 (2006) 326–348. | Zbl | DOI

O.R. Burggraf, Analytical and numerical studies of the structure of steady separated flows. J. Fluid Mech. 24 (1996) 113–151. | DOI

A.-L. Chaillou and M. Suri, Computable error estimators for the approximation of nonlinear problems by linearized models. Comput. Methods Appl. Mech. Eng. 196 (2006) 210–224. | Zbl | MR | DOI

A.-L. Chaillou and M. Suri, A posteriori estimation of the linearization error for strongly monotone nonlinear operators. Comput. Methods Appl. Mech. Eng. 205 (2007) 72–87. | Zbl | MR

A. El Akkad, A. El Khalfi and N. Guessous, 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

L. El Alaoui, A. Ern and M. Vohralík, 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

A. Ern and M. Vohralík, Adaptive inexact Newton methods with a posteriori stopping criteria for nonlinear diffusion PDEs. SIAMJ. Sci. Comput. 35 (2013) A1761–A1791. | Zbl | MR | DOI

E. Erturk, Discussions on driven cavity flow. Int. J. Numer. Meth. Fluids 60 (2009) 747–774. | Zbl | MR | DOI

E. Erturk, T.C. Corke and C. Gokcol, 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

F. Hecht, New development in FreeFem++. J. Numer. Math. 20 (2012) 251–266. | Zbl | MR | DOI

H. Jin and S. Prudhomme, A posteriori error estimation in finite element analysis. Comput. Methods Appl. Mech. Eng. 159 (1998) 19–48. | Zbl | MR

V. John, 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

M. Kawaguti, 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

J. Pousin and J. Rappaz, 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

S. Prudhomme and J.T. Oden, 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

R. Schreiber and H.B. Keller, 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

Cité par Sources :