[Stratégies de point fixe pour formulations variationnelles mixtes du problème stationnaire de Boussinesq]
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.
Accepté le :
Publié le :
Eligio Colmenares 1, 2 ; Gabriel N. Gatica 1, 2 ; Ricardo Oyarzúa 3, 2
@article{CRMATH_2016__354_1_57_0, author = {Eligio Colmenares and Gabriel N. Gatica and Ricardo Oyarz\'ua}, title = {Fixed point strategies for mixed variational formulations of the stationary {Boussinesq} problem}, journal = {Comptes Rendus. Math\'ematique}, pages = {57--62}, publisher = {Elsevier}, volume = {354}, number = {1}, year = {2016}, doi = {10.1016/j.crma.2015.10.004}, language = {en}, }
TY - JOUR AU - Eligio Colmenares AU - Gabriel N. Gatica AU - Ricardo Oyarzúa TI - Fixed point strategies for mixed variational formulations of the stationary Boussinesq problem JO - Comptes Rendus. Mathématique PY - 2016 SP - 57 EP - 62 VL - 354 IS - 1 PB - Elsevier DO - 10.1016/j.crma.2015.10.004 LA - en ID - CRMATH_2016__354_1_57_0 ER -
%0 Journal Article %A Eligio Colmenares %A Gabriel N. Gatica %A Ricardo Oyarzúa %T Fixed point strategies for mixed variational formulations of the stationary Boussinesq problem %J Comptes Rendus. Mathématique %D 2016 %P 57-62 %V 354 %N 1 %I Elsevier %R 10.1016/j.crma.2015.10.004 %G en %F CRMATH_2016__354_1_57_0
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] 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] Mixed and Hybrid Finite Element Methods, Springer Verlag, 1991
[3] 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] Analysis of an augmented mixed-primal formulation for the stationary Boussinesq problem, Numer. Methods Partial Differ. Equ. (2016), p. 22001 (in press) | DOI
[5] 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] A Simple Introduction to the Mixed Finite Element Method: Theory and Applications, Springer Briefs in Mathematics, Springer, Cham, 2014
[7] Dual mixed finite element methods for the Navier–Stokes equations, ESAIM: Math. Model. Numer. Anal., Volume 47 (2013), pp. 789-805
[8] An exactly divergence-free finite element method for a generalized Boussinesq problem, IMA J. Numer. Anal., Volume 34 (2014) no. 3, pp. 1104-1135
- 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
- Analysis of a new mixed FEM for stationary incompressible magneto-hydrodynamics, Computers Mathematics with Applications, Volume 127 (2022), p. 65 | DOI:10.1016/j.camwa.2022.09.017
- 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
- 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
- Analysis of an augmented fully-mixed finite element method for a bioconvective flows model, Journal of Computational and Applied Mathematics, Volume 393 (2021), p. 113504 | DOI:10.1016/j.cam.2021.113504
- 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
- A Fully-Mixed Formulation for the Steady Double-Diffusive Convection System Based upon Brinkman–Forchheimer Equations, Journal of Scientific Computing, Volume 85 (2020) no. 2 | DOI:10.1007/s10915-020-01305-x
- Lptheory for Boussinesq system with Dirichlet boundary conditions, Applicable Analysis, Volume 98 (2019) no. 1-2, p. 272 | DOI:10.1080/00036811.2018.1530762
- 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
- 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
- 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
- On
-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 - Analysis of an augmented fully-mixed formulation for the coupling of the Stokes and heat equations, ESAIM: Mathematical Modelling and Numerical Analysis, Volume 52 (2018) no. 5, p. 1947 | DOI:10.1051/m2an/2018027
- 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
- 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
- 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
- Dual-mixed finite element methods for the stationary Boussinesq problem, Computers Mathematics with Applications, Volume 72 (2016) no. 7, p. 1828 | DOI:10.1016/j.camwa.2016.08.011
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.
Commentaires - Politique
Vous devez vous connecter pour continuer.
S'authentifier