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Comptes Rendus Mathématique
Volume 354, n° 1
pages 57-62 (janvier 2016)
Doi : 10.1016/j.crma.2015.10.004
Received : 22 June 2015 ;  accepted : 1 October 2015
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
 

Eligio Colmenares c, a , Gabriel N. Gatica c, a , Ricardo Oyarzúa b, c
a Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile 
b GIMNAP-Departamento de Matemática, Universidad del Bío-Bío, Casilla 5-C, Concepción, Chile 
c CI2MA, Universidad de Concepción, Casilla 160-C, Concepción, Chile 

Abstract

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.

The full text of this article is available in PDF format.
Résumé

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.

The full text of this article is available in PDF format.

 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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