A posteriori analysis of the Chorin–Temam scheme for Stokes equations

Sébastien Boyaval; Marco Picasso

doi:10.1016/j.crma.2013.10.026

Numerical analysis

A posteriori analysis of the Chorin–Temam scheme for Stokes equations
[Analyse a posteriori du schéma Chorin–Temam pour les équations de Stokes]

Présenté par : Olivier Pironneau

Sébastien Boyaval ^1,² ; Marco Picasso ³

¹ Université Paris-Est, Laboratoire dʼhydraulique Saint-Venant (École nationale des ponts et chaussées – EDF R&D – CETMEF), 78401 Chatou cedex, France
² INRIA, MICMAC team project, Rocquencourt, France
³ MATHICSE, Station 8, École polytechnique fedérale de Lausanne, 1015 Lausanne, Switzerland

Comptes Rendus. Mathématique, Volume 351 (2013) no. 23-24, pp. 931-936.

Résumé
Abstract

On discrétise en temps, par le schéma Chorin–Temam, un problème de Stokes non stationnaire posé dans $D \subset R^{d}$ ( $d = 2, 3$ ), étant donnés $μ, f, u_{0}$ : $(P)$ trouver $(u, p)$ solution de $u |_{t = 0} = u_{0}$ , $u |_{\partial D} = 0$ et (1). En sʼinspirant des analyses de C. Bernardi et de R. Verfürth pour le schéma Euler rétrograde, nous construisons des estimateurs a posteriori pour lʼerreur commise sur $\nabla u$ en norme $L^{2} (0, T; L^{2} (D))$ . Notre étude est étayée par des expériences numériques.

We consider the Chorin–Temam scheme (the simplest pressure-correction projection method) for the time discretization of an unstationary Stokes problem in $D \subset R^{d}$ ( $d = 2, 3$ ) given $μ, f, u_{0}$ : $(P)$ find $(u, p)$ solution to $u |_{t = 0} = u_{0}$ , $u |_{\partial D} = 0$ and:

\frac{\partial u}{\partial t} - μ Δ u + \nabla p = f, div u = 0 on (0, T) \times D .

(1)

Inspired by the analyses of the Backward Euler scheme performed by C. Bernardi and R. Verfürth, we derive a posteriori estimators for the error on

\nabla u

L^{2} (0, T; L^{2} (D))

-norm. Our investigation is supported by numerical experiments.

Reçu le : 2013-04-23
Accepté le : 2013-10-22
Publié le : 2013-11-21

DOI : 10.1016/j.crma.2013.10.026

Affiliations des auteurs :

Sébastien Boyaval ^{1, 2} ; Marco Picasso ³

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     author = {S\'ebastien Boyaval and Marco Picasso},
     title = {A posteriori analysis of the {Chorin{\textendash}Temam} scheme for {Stokes} equations},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {931--936},
     publisher = {Elsevier},
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     year = {2013},
     doi = {10.1016/j.crma.2013.10.026},
     language = {en},
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Sébastien Boyaval; Marco Picasso. A posteriori analysis of the Chorin–Temam scheme for Stokes equations. Comptes Rendus. Mathématique, Volume 351 (2013) no. 23-24, pp. 931-936. doi : 10.1016/j.crma.2013.10.026. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2013.10.026/

Bibliographie
Cité par

[1] C. Bernardi; R. Verfürth A posteriori error analysis of the fully discretized time-dependent Stokes equations, M2AN Math. Model. Numer. Anal., Volume 38 (2004), pp. 437-455

[2] A.-J. Chorin Numerical solution of the Navier–Stokes equations, Math. Comput., Volume 22 (1968), pp. 754-762

[3] V. Girault; P.-A. Raviart Finite Element Approximation of the Navier–Stokes Equations, Lecture Notes in Mathematics, vol. 749, Springer-Verlag, Berlin, 1979

[4] J.-L. Guermond Some implementations of projection methods for Navier–Stokes equations, RAIRO Modél. Math. Anal. Numér., Volume 30 (1996), pp. 637-667

[5] J.L. Guermond; P. Minev; J. Shen An overview of projection methods for incompressible flows, Comput. Methods Appl. Mech. Eng., Volume 195 (2006), pp. 6011-6045

[6] J.-L. Guermond; L. Quartapelle On stability and convergence of projection methods based on pressure Poisson equation, Int. J. Numer. Methods Fluids, Volume 26 (1998), pp. 1039-1053

[7] N. Kharrat; Z. Mghazli Time error estimators for the Chorin–Temam scheme, ARIMA, Volume 13 (2010), pp. 33-46

[8] N. Kharrat; Z. Mghazli A posteriori error analysis of time-dependent Stokes problem by Chorin–Temam scheme, Calcolo (2011), pp. 1-21 | DOI

[9] A. Prohl Projection and Quasi-Compressibility Methods for Solving the Incompressible Navier–Stokes Equations, Advances in Numerical Mathematics, B.G. Teubner GmbH, Stuttgart, 1997

[10] R. Temam Une méthode dʼapproximation de la solution des équations de Navier–Stokes, Bull. Soc. Math. Fr., Volume 96 (1968), pp. 115-152

[11] R. Temam Navier–Stokes Equations, Studies in Mathematics and Its Applications, vol. 2, North-Holland Publishing Co., Amsterdam, 1979

[12] R. Verfürth A posteriori error analysis of space–time finite element discretizations of the time-dependent Stokes equations, Calcolo, Volume 47 (2010), pp. 149-167

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