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

Presented by: Olivier Pironneau

Sébastien Boyaval ^1,²; Marco Picasso ³

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

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

Abstract (Original language)
Résumé

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

in

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

-norm. Our investigation is supported by numerical experiments.

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.

Received: 2013-04-23
Accepted: 2013-10-22
Published online: 2013-11-21

DOI: 10.1016/j.crma.2013.10.026

Author's affiliations:

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

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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},
     year = {2013},
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     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

References
Cited by

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