Comptes Rendus
Partial differential equations/Numerical analysis
Fixed point strategies for mixed variational formulations of the stationary Boussinesq problem
[Stratégies de point fixe pour formulations variationnelles mixtes du problème stationnaire de Boussinesq]
Comptes Rendus. Mathématique, Volume 354 (2016) no. 1, pp. 57-62.

Dans cet article, on présente les principaux résultats concernant l'analyse de résolution de deux nouvelles formulations variationnelles mixtes pour le problème stationnaire de Boussinesq. Plus précisément, on introduit des approches mixtes-primal et entièrement mixtes, toute les deux convenablement augmentées avec des équations de type Galerkin, et l'on montre que les régimes qui en résultent peuvent être réécrits, de maniére équivalente, comme équations d'opérateur de point fixe. Ainsi, les arguments classiques de l'analyse fonctionnelle linéaires et non linéaires sont utilisés pour conclure qu'elles sont bien posées.

In this paper, we report on the main results concerning the solvability analysis of two new mixed variational formulations for the stationary Boussinesq problem. More precisely, we introduce mixed-primal and fully-mixed approaches, both of them suitably augmented with Galerkin-type equations, and show that the resulting schemes can be rewritten, equivalently, as fixed-point operator equations. Then, classical arguments from linear and nonlinear functional analysis are employed to conclude that they are well-posed.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2015.10.004

Eligio Colmenares 1, 2 ; Gabriel N. Gatica 1, 2 ; Ricardo Oyarzúa 3, 2

1 Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile
2 CI
3 GIMNAP-Departamento de Matemática, Universidad del Bío-Bío, Casilla 5-C, Concepción, Chile
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Eligio Colmenares; Gabriel N. Gatica; Ricardo Oyarzúa. Fixed point strategies for mixed variational formulations of the stationary Boussinesq problem. Comptes Rendus. Mathématique, Volume 354 (2016) no. 1, pp. 57-62. doi : 10.1016/j.crma.2015.10.004. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2015.10.004/

[1] C. Bernardi; B. Métivet; B. Pernaud-Thomas Couplage des équations de Navier–Stokes et de la chaleur: le modèle et son approximation par éléments finis, ESAIM: Math. Model. Numer. Anal., Volume 29 (1995) no. 7, pp. 871-921

[2] F. Brezzi; M. Fortin Mixed and Hybrid Finite Element Methods, Springer Verlag, 1991

[3] E. Colmenares; G.N. Gatica; R. Oyarzúa An augmented fully-mixed finite element method for the stationary Boussinesq problem, Centro de Investigación en Ingeniería Matemática (CI, 2015 http://www.ci2ma.udec.cl/publicaciones/prepublicaciones (Preprint 2015-30 available at)

[4] E. Colmenares; G.N. Gatica; R. Oyarzúa Analysis of an augmented mixed-primal formulation for the stationary Boussinesq problem, Numer. Methods Partial Differ. Equ. (2016), p. 22001 (in press) | DOI

[5] M. Farhloul; S. Nicaise; L. Paquet A refined mixed finite element method for the Boussinesq equations in polygonal domains, IMA J. Numer. Anal., Volume 21 (2001) no. 2, pp. 525-551

[6] G.N. Gatica A Simple Introduction to the Mixed Finite Element Method: Theory and Applications, Springer Briefs in Mathematics, Springer, Cham, 2014

[7] J.S. Howell; N. Walkington Dual mixed finite element methods for the Navier–Stokes equations, ESAIM: Math. Model. Numer. Anal., Volume 47 (2013), pp. 789-805

[8] R. Oyarzúa; T. Qin; D. Schötzau An exactly divergence-free finite element method for a generalized Boussinesq problem, IMA J. Numer. Anal., Volume 34 (2014) no. 3, pp. 1104-1135

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Cité par 17 documents. Sources : Crossref

This work was partially supported by CONICYT-Chile through BASAL project CMM, Universidad de Chile, project Anillo ACT1118 (ANANUM), and project Fondecyt 11121347; by Centro de Investigación en Ingeniería Matemática (CI2MA), Universidad de Concepción; and by Universidad del Bío-Bío through DIUBB project 120808 GI/EF.

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