Stability estimates for solving Stokes problem with nonconforming finite elements

Erell Jamelot

doi:10.5802/crmath.707

Article de recherche - Analyse numérique

Stability estimates for solving Stokes problem with nonconforming finite elements
[Estimations de stabilité pour résoudre le problème de Stokes avec des éléments finis non conformes]

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.

Résumé
Abstract

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 $H (div)$ pour certaines expériences numériques.

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.

Reçu le : 2023-04-07
Révisé le : 2024-12-03
Accepté le : 2025-02-02
Publié le : 2025-03-25

DOI : 10.5802/crmath.707

Classification : 65N30, 35J57, 76D07

Affiliations des auteurs :

Erell Jamelot ¹

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

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@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},
}

TY  - JOUR
AU  - Erell Jamelot
TI  - Stability estimates for solving Stokes problem with nonconforming finite elements
JO  - Comptes Rendus. Mathématique
PY  - 2025
SP  - 115
EP  - 137
VL  - 363
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.707
LA  - en
ID  - CRMATH_2025__363_G2_115_0
ER  -

%0 Journal Article
%A Erell Jamelot
%T Stability estimates for solving Stokes problem with nonconforming finite elements
%J Comptes Rendus. Mathématique
%D 2025
%P 115-137
%V 363
%I Académie des sciences, Paris
%R 10.5802/crmath.707
%G en
%F CRMATH_2025__363_G2_115_0

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/

Bibliographie
Cité par

[1] Mark Ainsworth; Alejandro Allendes; Gabriel R. Barrenechea; Richard Rankin Computable error bounds for nonconforming Fortin–Soulie finite element approximation of the Stokes problem, IMA J. Numer. Anal., Volume 32 (2011) no. 2, pp. 417-447 | DOI | MR | Zbl

[2] Thomas Apel; Volker Kempf; Alexander Linke; Christian Merdon A nonconforming pressure-robust finite element method for the Stokes equations on anisotropic meshes, IMA J. Numer. Anal., Volume 42 (2022) no. 1, pp. 392-416 | DOI | MR | Zbl

[3] Thomas Apel; Serge Nicaise; Joachim Schöberl Crouzeix-Raviart type finite elements on anisotropic meshes, Numer. Math., Volume 89 (2001) no. 2, pp. 193-223 | DOI | MR | Zbl

[4] Ágnes Baran; Gisbert Stoyan Gauss-Legendre elements: a stable, higher order non-conforming finite element family, Computing, Volume 79 (2007) no. 1, pp. 1-21 | DOI | MR | Zbl

[5] Mathieu Barré; Patrick Ciarlet The $T$ -coercivity approach for mixed problems, C. R., Math., Acad. Sci. Paris, Volume 362 (2024), pp. 1051-1088 | DOI | MR | Zbl

[6] Christine Bernardi; Martin Costabel; Monique Dauge; Vivette Girault Continuity properties of the inf-sup constant for the divergence, SIAM J. Math. Anal., Volume 48 (2016) no. 2, pp. 1250-1271 | DOI | MR | Zbl

[7] Christine Bernardi; Frédéric Hecht More pressure in the finite element discretization of the Stokes problem, ESAIM, Math. Model. Numer. Anal., Volume 34 (2000) no. 5, pp. 953-980 | DOI | Numdam | MR | Zbl

[8] Daniele Boffi; Franco Brezzi; Michel Fortin Mixed finite element methods and applications, Springer Series in Computational Mathematics, 44, Springer, 2013, xiv+685 pages | DOI | MR

[9] Anne-Sophie Bonnet-Ben Dhia; Patrick Ciarlet Notes de cours sur les méthodes variationnelles pour l’analyse de problèmes non coercifs (2024) M.Sc. AMS lecture notes (ENSTA-IPP) | HAL

[10] Christian Brennecke; Alexander Linke; Christian Merdon; Joachim Schöberl Optimal and pressure independent $L^{2}$ velocity error estimates for a modified Crouzeix-Raviart Stokes element with BDM reconstructions, J. Comput. Math., Volume 33 (2015) no. 2, pp. 191-208 | DOI | MR

[11] Susanne Cecelia Brenner Poincaré–Friedrichs inequalities for piecewise $H^{1}$ functions, SIAM J. Numer. Anal., Volume 41 (2003) no. 1, pp. 306-324 | DOI | MR | Zbl

[12] Franco Brezzi; Jim Douglas; Luisa Donatella Marini Two families of mixed finite elements for second order elliptic problems, Numer. Math., Volume 47 (1985) no. 2, pp. 217-235 | DOI | MR | Zbl

[13] Erik Burman; Peter Hansbo Stabilized Crouzeix-Raviart element for the Darcy-Stokes problem, Numer. Methods Partial Differ. Equations, Volume 21 (2005) no. 5, pp. 986-997 | DOI | MR | Zbl

[14] Carsten Carstensen; Stefan Sauter Critical functions and inf-sup stability of Crouzeix-Raviart elements, Comput. Math. Appl., Volume 108 (2022), pp. 12-23 | DOI | MR | Zbl

[15] Carsten Carstensen; Stefan Sauter Crouzeix-Raviart triangular elements are inf-sup stable, Math. Comput., Volume 91 (2022) no. 337, pp. 2041-2057 | DOI | MR | Zbl

[16] Lucas Chesnel; Patrick Ciarlet $T$ -coercivity and continuous Galerkin methods: application to transmission problems with sign changing coefficients, Numer. Math., Volume 124 (2013), pp. 1-29 | DOI | MR | Zbl

[17] Patrick Ciarlet T-coercivity: Application to the discretization of Helmhotz-like problems, Comput. Math. Appl., Volume 64 (2012) no. 1, pp. 22-24 | DOI | MR | Zbl

[18] Patrick Ciarlet Analysis of the Scott–Zhang interpolation in the fractional order Sobolev spaces, J. Numer. Math., Volume 21 (2013) no. 3, pp. 173-180 | MR | Zbl

[19] Patrick Ciarlet; Erell Jamelot Explicit T-coercivity for the Stokes problem: a coercive finite element discretization (2025) (To appear in Comput. Math. Appl.) | HAL

[20] Patrick Ciarlet; Erell Jamelot; Félix D. Kpadonou Domain decomposition methods for the diffusion equation with low-regularity solution, Comput. Math. Appl., Volume 74 (2017) no. 10, pp. 2369-2384 | DOI | MR | Zbl

[21] Martin Costabel; Monique Dauge On the inequalities of Babuška-Aziz, Friedrichs and Horgan-Payne, Arch. Ration. Mech. Anal., Volume 217 (2015), pp. 873-898 | DOI | MR | Zbl

[22] Michel Crouzeix; Pierre-Arnaud Raviart Conforming and nonconforming finite element methods for solving the stationary Stokes equations, Rev. Franç. Autom. Inform. Rech. Opérat., R, Volume 7 (1973) no. 3, pp. 33-75 | Numdam | MR

[23] Enzo A. Dari; Ricardo G. Durán; Claudio Padra Error estimators for nonconforming finite element approximations of the Stokes problem, Math. Comput., Volume 64 (1995) no. 211, pp. 1017-1033 | DOI | MR | Zbl

[24] Lars Diening; Johannes Storn; Tabea Tscherpel Fortin operator for Taylor-Hood element, Numer. Math., Volume 150 (2022) no. 2, pp. 671-689 | DOI | MR | Zbl

[25] Willy Dörfler; Mark Ainsworth Reliable a posteriori error control for nonconforming finite element approximation of Stokes flow, Math. Comput., Volume 74 (2005) no. 252, pp. 1599-1619 | DOI | MR | Zbl

[26] Alexandre Ern; Jean-Luc Guermond Finite elements I–Approximation and interpolation, Texts in Applied Mathematics, 72, Springer, 2021, xii+325 pages | DOI | MR

[27] Alexandre Ern; Jean-Luc Guermond Finite elements II–Galerkin approximation, elliptic and mixed PDEs, Texts in Applied Mathematics, 73, Springer, 2021, ix+492 pages | DOI | MR

[28] Maurice S. Fabien; Johnny Guzmán; Michael Neilan; Ahmed Zytoon Low-order divergence-free approximations for the Stokes problem on Worsey–Farin and Powell–Sabin splits, Comput. Methods Appl. Mech. Eng., Volume 390 (2022), 114444, 21 pages | MR | Zbl

[29] Michel Fortin; M. Soulie A non-conforming piecewise quadratic finite element on triangles, Int. J. Numer. Methods Eng., Volume 19 (1983) no. 4, pp. 505-520 | DOI | MR | Zbl

[30] Dietmar Gallistl Rayleigh-Ritz approximation of the inf-sup constant for the divergence, Math. Comput., Volume 88 (2019) no. 315, pp. 73-89 | DOI | MR | Zbl

[31] Gabriel N. Gatica A simple introduction to the mixed finite element method. Theory and applications, SpringerBriefs in Mathematics, Springer, 2014, xii+132 pages | DOI | MR

[32] Vivette Girault; Pierre-Arnaud Raviart Finite element methods for Navier-Stokes equations, Springer Series in Computational Mathematics, 5, Springer, 1986, x+374 pages | DOI | MR

[33] Léandre Giret Non-Conforming Domain Decomposition for the Multigroup Neutron SPN Equation, Ph. D. Thesis, Université Paris-Saclay (France) (2018)

[34] Frédéric Hecht Construction d’une base de fonctions $P_{1}$ non conforme à divergence nulle dans $ℝ^{3}$ , RAIRO, Anal. Numér., Volume 15 (1981) no. 2, pp. 119-150 | DOI | MR | Zbl

[35] Erell Jamelot Nonconforming mixed finite elements code to solve Stokes Problem, $2 D$ , $k = 1, 2$ (2022) https://github.com/cea-trust-platform/stokes_ncfem

[36] Erell Jamelot; Patrick Ciarlet Fast non-overlapping Schwarz domain decomposition methods for solving the neutron diffusion equation, J. Comput. Phys., Volume 241 (2013), pp. 445-463 | DOI | MR | Zbl

[37] Erell Jamelot; Patrick Ciarlet; Stefan Sauter Stability of the $P_{n c}^{1} - (P^{0} + P^{1})$ element (2025) (To appear in the ENUMATH 2023 proceedings) | HAL

[38] Alexander Linke On the Role of the Helmholtz-Decomposition in Mixed Methods for Incompressible Flows and a New Variational Crime, Comput. Methods Appl. Mech. Eng., Volume 268 (2014), pp. 782-800 | DOI | MR | Zbl

[39] Pierre-Arnaud Raviart; Jean-Marie Thomas A mixed finite element method for second order elliptic problems, Mathematical aspects of finite element methods (Lecture Notes in Mathematics), Volume 606, Springer (1977), pp. 292-315 | DOI | Zbl

[40] Stefan Sauter The inf-sup constant for $h p$ -Crouzeix-Raviart triangular elements (2022) | arXiv

[41] Stefan Sauter The inf-sup constant for $h p$ -Crouzeix-Raviart triangular elements, Comput. Math. Appl., Volume 149 (2023), pp. 49-70 | DOI | MR | Zbl

[42] Stefan Sauter; Céline Torres On the Inf-Sup Stabillity of Crouzeix-Raviart Stokes Elements in 3D, Math. Comput., Volume 92 (2023), pp. 1033-1059 | DOI | MR | Zbl

[43] Larkin Ridgway Scott; Michael Steenstrup Vogelius Norm estimates for a maximal right inverse of the divergence operator in spaces of piecewise polynomials, RAIRO, Modélisation Math. Anal. Numér., Volume 19 (1985) no. 1, pp. 111-143 | DOI | Numdam | MR | Zbl

[44] Larkin Ridgway Scott; Shangyou Zhang Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comput., Volume 54 (1990), pp. 483-493 | DOI | MR | Zbl

[45] Rolf Stenberg Error analysis of some finite element methods for the Stokes problem, Math. Comput., Volume 54 (1990), pp. 495-508 | DOI | MR | Zbl

[46] Luc Tartar An introduction to Sobolev spaces and interpolation spaces, Lecture Notes of the Unione Matematica Italiana, 3, Springer, 2007, xxvi+218 pages | MR

[47] Cedric Taylor; P. Hood A numerical solution of the Navier-Stokes equations using the finite element technique, Comput. Fluids, Volume 1 (1973), pp. 73-100 | DOI | Zbl

[48] Martin Vohralík On the discrete Poincaré-Friedrichs inequlities for nonconforming approximations of the Sobolev space $H^{1}$ , Numer. Funct. Anal. Optim., Volume 26 (2005), pp. 925-952 | DOI | MR | Zbl

[49] Shangyou Zhang A new family of stable mixed finite elements for the 3D Stokes equations, Math. Comput., Volume 74 (2005), pp. 543-554 | DOI | MR | Zbl

Cité par Sources :

Commentaires - Politique