Fixed point strategies for mixed variational formulations of the stationary Boussinesq problem

Eligio Colmenares; Gabriel N. Gatica; Ricardo Oyarzúa

doi:10.1016/j.crma.2015.10.004

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]

Présenté par : Olivier Pironneau

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

Comptes Rendus. Mathématique, Volume 354 (2016) no. 1, pp. 57-62.

Résumé
Abstract

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 : 2015-06-22
Accepté le : 2015-10-01
Publié le : 2015-11-02

DOI : 10.1016/j.crma.2015.10.004

Affiliations des auteurs :

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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     title = {Fixed point strategies for mixed variational formulations of the stationary {Boussinesq} problem},
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     pages = {57--62},
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     doi = {10.1016/j.crma.2015.10.004},
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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/

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Mario Álvarez; Eligio Colmenares; Filánder A. Sequeira Analysis of a semi-augmented mixed finite element method for double-diffusive natural convection in porous media, Computers Mathematics with Applications, Volume 114 (2022), p. 112 | DOI:10.1016/j.camwa.2022.03.032
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Sergio Caucao; Gabriel N. Gatica; Ricardo Oyarzúa; Paulo Zúñiga A Posteriori Error Analysis of a Mixed Finite Element Method for the Coupled Brinkman–Forchheimer and Double-Diffusion Equations, Journal of Scientific Computing, Volume 93 (2022) no. 2 | DOI:10.1007/s10915-022-02010-7
Sergio Caucao; Gabriel N. Gatica; Juan P. Ortega A fully-mixed formulation in Banach spaces for the coupling of the steady Brinkman–Forchheimer and double-diffusion equations, ESAIM: Mathematical Modelling and Numerical Analysis, Volume 55 (2021) no. 6, p. 2725 | DOI:10.1051/m2an/2021072
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Sergio Caucao; Ricardo Oyarzúa; Segundo Villa-Fuentes A new mixed-FEM for steady-state natural convection models allowing conservation of momentum and thermal energy, Calcolo, Volume 57 (2020) no. 4 | DOI:10.1007/s10092-020-00385-3
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Paul Acevedo; Chérif Amrouche; Carlos Conca Lptheory for Boussinesq system with Dirichlet boundary conditions, Applicable Analysis, Volume 98 (2019) no. 1-2, p. 272 | DOI:10.1080/00036811.2018.1530762
Eligio Colmenares; Gabriel N. Gatica; Ricardo Oyarzúa A posteriori error analysis of an augmented fully-mixed formulation for the stationary Boussinesq model, Computers Mathematics with Applications, Volume 77 (2019) no. 3, p. 693 | DOI:10.1016/j.camwa.2018.10.009
James Woodfield; Mario Alvarez; Bryan Gómez-Vargas; Ricardo Ruiz-Baier Stability and finite element approximation of phase change models for natural convection in porous media, Journal of Computational and Applied Mathematics, Volume 360 (2019), p. 117 | DOI:10.1016/j.cam.2019.04.003
Sergio Caucao; Gabriel N. Gatica; Ricardo Oyarzúa A posteriori error analysis of an augmented fully mixed formulation for the nonisothermal Oldroyd–Stokes problem, Numerical Methods for Partial Differential Equations, Volume 35 (2019) no. 1, p. 295 | DOI:10.1002/num.22301
Raimund Bürger; Paul E. Méndez; Ricardo Ruiz-Baier On $H (d i v)$ -conforming Methods for Double-diffusion Equations in Porous Media, SIAM Journal on Numerical Analysis, Volume 57 (2019) no. 3, p. 1318 | DOI:10.1137/18m1196108
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Gabriel N. Gatica; Mauricio Munar; Filánder A. Sequeira A mixed virtual element method for the Navier–Stokes equations, Mathematical Models and Methods in Applied Sciences, Volume 28 (2018) no. 14, p. 2719 | DOI:10.1142/s0218202518500598
Eligio Colmenares; Gabriel N. Gatica; Ricardo Oyarzúa A posteriori error analysis of an augmented mixed-primal formulation for the stationary Boussinesq model, Calcolo, Volume 54 (2017) no. 3, p. 1055 | DOI:10.1007/s10092-017-0219-2
Marco Discacciati; Ricardo Oyarzúa A conforming mixed finite element method for the Navier–Stokes/Darcy coupled problem, Numerische Mathematik, Volume 135 (2017) no. 2, p. 571 | DOI:10.1007/s00211-016-0811-4
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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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