Stability estimates for solving Stokes problem with nonconforming finite elements

Erell Jamelot

doi:10.5802/crmath.707

Research article - Numerical analysis

Stability estimates for solving Stokes problem with nonconforming finite elements

Erell Jamelot ¹

¹ Université Paris-Saclay, CEA, Service de Thermo-hydraulique et de Mécanique des Fluides, 91191 Gif-sur-Yvette, France

Comptes Rendus. Mathématique, Volume 363 (2025), pp. 115-137.

Abstract (OV)
Résumé

We propose to analyse the discretization of the Stokes problem with nonconforming finite elements in light of the T-coercivity. First we exhibit a family of operators to prove T-coercivity and we show that the stability constant is equal to the classical one up to a constant which depends on the Babuška–Aziz constant. Then we explicit the stability constants with respect to the shape regularity parameter for order 1 in 2 or 3 dimensions, and order 2 in 2 dimensions. In this last case, we improve the result of the original Fortin–Soulie paper. Second, we illustrate the importance of using a divergence-free velocity reconstruction on some numerical experiments.

Nous proposons d’analyser la discrétisation du problème de Stokes avec des éléments finis non conformes à la lumière de la T-coercivité. Tout d’abord, pour prouver la T-coercivité, nous exhibons une famille d’opérateurs et nous montrons que la constante de stabilité est égale à la constante de stabilité classique, à une constante près qui dépend de la constante de Babuška–Aziz. Par la suite, nous explicitons les constantes de stabilité par rapport au paramètre de régularité de forme pour l’ordre $1$ en dimension $2$ ou $3$, et l’ordre $2$ en dimension $2$. Dans ce dernier cas, nous améliorons le résultat de l’article original de Fortin–Soulie. Ensuite nous illustrons l’importance d’utiliser une méthode de projection conforme dans $\mathbf{H}(\mathrm{div})$ pour certaines expériences numériques.

Received: 2023-04-07
Revised: 2024-12-03
Accepted: 2025-02-02
Published online: 2025-03-25

DOI: 10.5802/crmath.707

Classification: 65N30, 35J57, 76D07

Author's affiliations:

Erell Jamelot ¹

¹ Université Paris-Saclay, CEA, Service de Thermo-hydraulique et de Mécanique des Fluides, 91191 Gif-sur-Yvette, France

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

@article{CRMATH_2025__363_G2_115_0,
     author = {Erell Jamelot},
     title = {Stability estimates for solving {Stokes} problem with nonconforming finite elements},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {115--137},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {363},
     year = {2025},
     doi = {10.5802/crmath.707},
     language = {en},
}

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Erell Jamelot. Stability estimates for solving Stokes problem with nonconforming finite elements. Comptes Rendus. Mathématique, Volume 363 (2025), pp. 115-137. doi : 10.5802/crmath.707. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.707/

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