A three field stabilized finite element method for the Stokes equations

Mohamed Amara; Eliseo Chacón Vera; David Trujillo

doi:10.1016/S1631-073X(02)02319-1

A three field stabilized finite element method for the Stokes equations
[Une méthode d'éléments finis stabilisée à trois champs pour les équations de Stokes]

Mohamed Amara ¹ ; Eliseo Chacón Vera ² ; David Trujillo ¹

¹ IPRA-LMA, Université de Pau et des Pay de l'Adour, 64000 Pau, France
² Departamento de ecuaciones diferenciales y analysis, Universidad de Sevilla, 41080 Sevilla, Spain

Comptes Rendus. Mathématique, Volume 334 (2002) no. 7, pp. 603-608.

Résumé
Abstract

On propose dans ce travail, une formulation vitesse-tourbillon-pression pour le problème de Stokes bidimensionnel dans lequel on impose des conditions au bord non standard. On s'intèresse plus précisément aux cas où, sur certaines parties du bord, sont données la pression et la composante tangentielle de la vitesse ou bien le tourbillon et la composante normale de la vitesse. En partant d'une formulation mixte variationnelle le problème est résolu avec des hypothèses minimales sur la régularité. Dans cette formulation, les inconnues principales sont la pression et le tourbillon, tandis que la vitesse joue le rôle du multiplicateur. Nous présentons le problème discrétisé associé, pour lequel nous rajoutons un terme de stabilisation. Un résultat de convergence, décrivant le comportement de l'erreur d'approximation a priori, est démontré. Nous terminons par quelques résultats numériques.

We consider in this work the boundary value problem for Stokes equations on a two dimensional domain in cases where non-standard boundary conditions are given. We study the cases where pressure and normal or tangential components of the velocity are given in different parts of the boundary and solve the problem with a minimal regularity. We introduce the problem and its variational formulation which is a mixed one. The principal unknowns are the pressure and the vorticity, the multiplier is the velocity. We present the numerical discretization which needs some stabilization. We prove the convergence and the behavior of the a priori error estimates. Some numerical tests are also presented.

Reçu le : 2001-05-28
Révisé le : 2002-02-05
Publié le : 2002-01-01

DOI : 10.1016/S1631-073X(02)02319-1

Affiliations des auteurs :

Mohamed Amara ¹ ; Eliseo Chacón Vera ² ; David Trujillo ¹

¹ IPRA-LMA, Université de Pau et des Pay de l'Adour, 64000 Pau, France
² Departamento de ecuaciones diferenciales y analysis, Universidad de Sevilla, 41080 Sevilla, Spain

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     title = {A three field stabilized finite element method for the {Stokes} equations},
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     language = {en},
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Mohamed Amara; Eliseo Chacón Vera; David Trujillo. A three field stabilized finite element method for the Stokes equations. Comptes Rendus. Mathématique, Volume 334 (2002) no. 7, pp. 603-608. doi : 10.1016/S1631-073X(02)02319-1. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)02319-1/

Bibliographie
Cité par

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Huoyuan Duan; Roger C. E. Tan; Duowei Zhu A pressure-residual augmented GLS stabilized method for a type of Stokes equations with nonstandard boundary conditions, Advances in Computational Mathematics, Volume 50 (2024) no. 5 | DOI:10.1007/s10444-024-10204-w
Tujin Kim; Daomin Cao Fluid Equations, Equations of Motion for Incompressible Viscous Fluids (2021), p. 41 | DOI:10.1007/978-3-030-78659-5_2
Tujin Kim Regularity of solutions to the Navier-Stokes equations with a nonstandard boundary condition, Acta Mathematicae Applicatae Sinica, English Series, Volume 31 (2015) no. 3, p. 707 | DOI:10.1007/s10255-015-0497-x
Tujin Kim; Daomin Cao Some properties on the surfaces of vector fields and its application to the Stokes and Navier–Stokes problems with mixed boundary conditions, Nonlinear Analysis: Theory, Methods Applications, Volume 113 (2015), p. 94 | DOI:10.1016/j.na.2014.09.017
Tujin Kim; Daomin Cao Local Exact Controllability of the Navier–Stokes Equations with the Condition on the Pressure on Parts of the Boundary, SIAM Journal on Control and Optimization, Volume 48 (2010) no. 6, p. 3805 | DOI:10.1137/060650143

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