[1]
|
R. A. Adams and J. J. F. Fournier, Sobolev Spaces, Second edition, Pure and Applied Mathematics (Amsterdam), 140. Elsevier/Academic Press, Amsterdam, 2003.
|
[2]
|
R. Aldbaissy, F. Hecht, G. Mansour and T. Sayah, A full discretisation of the time-dependent Boussinesq (buoyancy) model with nonlinear viscosity, Calcolo, 55 (2018), Art. 44, 49 pp.
doi: 10.1007/s10092-018-0285-0.
|
[3]
|
A. Allendes, G. R. Barrenechea and C. Naranjo, A divergence-free low-order stabilized finite element method for a generalized steady state Boussinesq problem, Comput. Methods Appl. Mech. Engrg., 340 (2018), 90-120.
doi: 10.1016/j.cma.2018.05.020.
|
[4]
|
J. A. Almonacid and G. N. Gatica, A fully-mixed finite element method for the $n$-dimensional Boussinesq problem with temperature-dependent parameters, Comput. Methods Appl. Math., 20 (2020), 187-213.
doi: 10.1515/cmam-2018-0187.
|
[5]
|
J. A. Almonacid, G. N. Gatica and R. Oyarzúa, A mixed-primal finite element method for the Boussinesq problem with temperature-dependent viscosity, Calcolo, 55 (2018), Art. 36, 42 pp.
doi: 10.1007/s10092-018-0278-z.
|
[6]
|
M. S Alnæs, J. Blechta, J. Hake, A. Johansson, B. Kehlet, A. Logg, C. Richardson, J. Ring, M. E. Rognes and G. N. Wells, The FEniCS project version 1.5, Arch. Numer. Softw., 3 (2015), 9-23.
|
[7]
|
M. Alvarez, G. N. Gatica, B. Gómez-Vargas and R. Ruiz-Baier, New mixed finite element methods for natural convection with phase-change in porous media, J. Sci. Comput., 80 (2019), 141-174.
doi: 10.1007/s10915-019-00931-4.
|
[8]
|
M. Alvarez, G. N. Gatica and R. Ruiz-Baier, An augmented mixed-primal finite element method for a coupled flow-transport problem, ESAIM Math. Model. Numer. Anal., 49 (2015), 1399-1427.
doi: 10.1051/m2an/2015015.
|
[9]
|
M. Alvarez, G. N. Gatica and R. Ruiz-Baier, A mixed-primal finite element approximation of a sedimentation-consolidation system, Math. Models Methods Appl. Sci., 26 (2016), 867-900.
doi: 10.1142/S0218202516500202.
|
[10]
|
P. R. Amestoy, I. S. Duff and J.-Y. L'Excellent, Multifrontal parallel distributed symmetric and unsymmetric solvers, Comput. Methods Appl. Mech. Engrg., 184 (2000), 501-520.
doi: 10.1016/S0045-7825(99)00242-X.
|
[11]
|
C. Bernardi, B. Métivet and B. Pernaud-Thomas, Couplage des équations de Navier-Stokes et de la chaleur: Le modèle et son approximation par éléments finis, RAIRO Modél. Math. Anal. Numér., 29 (1995), 871-921.
doi: 10.1051/m2an/1995290708711.
|
[12]
|
B. Blankenbach, F. Busse, U. Christensen, L. Cserepes, D. Gunkel, U. Hansen, H. Harder, G. Jarvis, M. Koch, G. Marquart, D. Moore, P. Olson, H. Schmeling and T. Schnaubelt, A benchmark comparison for mantle convection codes, Geophys. J. Int., 98 (1989), 23-38.
doi: 10.1111/j.1365-246X.1989.tb05511.x.
|
[13]
|
J. Boland and W. Layton, An analysis of the FEM for natural convection problems, Numer. Methods Partial Differential Equations, 6 (1990), 115-126.
doi: 10.1002/num.1690060202.
|
[14]
|
F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer Series in Computational Mathematics, 15. Springer-Verlag, New York, 1991.
doi: 10.1007/978-1-4612-3172-1.
|
[15]
|
J. Camaño, G. N. Gatica, R. Oyarzúa and R. Ruiz-Baier, An augmented stress-based mixed finite element method for the Navier-Stokes equations with nonlinear viscosity, Numer. Methods Partial Differential Equations, 33 (2017), 1692-1725.
doi: 10.1002/num.22166.
|
[16]
|
J. Camaño, R. Oyarzúa, R. Ruiz-Baier and G. Tierra, Error analysis of an augmented mixed method for the Navier-Stokes problem with mixed boundary conditions, IMA J. Numer. Anal., 38 (2018), 1452-1484.
doi: 10.1093/imanum/drx039.
|
[17]
|
S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, The International Series of Monographs on Physics Clarendon Press, Oxford, 1961.
|
[18]
|
Y. Y. Chen, B. W. Li and J. K. Zhang, Spectral collocation method for natural convection in a square porous cavity with local thermal equilibrium and non-equilibrium models, Int. J. Heat Mass Transfer, 64 (2013), 35-49.
doi: 10.1016/j.ijheatmasstransfer.2016.01.007.
|
[19]
|
A. Çibik and S. Kaya, A projection-based stabilized finite element method for steady-state natural convection problem, J. Math. Anal. Appl., 381 (2011), 469-484.
doi: 10.1016/j.jmaa.2011.02.020.
|
[20]
|
E. Colmenares, G. N. Gatica and R. Oyarzúa, Analysis of an augmented mixed-primal formulation for the stationary Boussinesq problem, Numer. Methods Partial Differential Equations, 32 (2016), 445-478.
doi: 10.1002/num.22001.
|
[21]
|
E. Colmenares and M. Neilan, Dual-mixed finite element methods for the stationary Boussinesq problem, Comput. Math. Appl., 72 (2016), 1828-1850.
doi: 10.1016/j.camwa.2016.08.011.
|
[22]
|
A. Dalal and M. K. Das, Natural convection in a rectangular cavity heated from below and uniformly cooled from the top and both sides, Numer. Heat Tr. A-Appl., 49 (2006), 301-322.
doi: 10.1080/10407780500343749.
|
[23]
|
H. Dallmann and D. Arndt, Stabilized finite element methods for the Oberbeck-Boussinesq model, J. Sci. Comput., 69 (2016), 244-273.
doi: 10.1007/s10915-016-0191-z.
|
[24]
|
M. Farhoul, S. Nicaise and L. Paquet, A mixed formulation of Boussinesq equations: Analysis of nonsingular solutions, Math. Comp., 69 (2000), 965-986.
doi: 10.1090/S0025-5718-00-01186-8.
|
[25]
|
M. Farhloul, S. Nicaise and L. Paquet, A refined mixed finite element method for the Boussinesq equations in polygonal domains, IMA J. Numer. Anal., 21 (2001), 525-551.
doi: 10.1093/imanum/21.2.525.
|
[26]
|
E. Feireisl and J. Málek, On the Navier-Stokes equations with temperature-dependent transport coefficients, Int. J. Diff. Eqns., 2006 (2006), Art. ID 90616, 14 pp.
doi: 10.1155/denm/2006/90616.
|
[27]
|
G. N. Gatica, An augmented mixed finite element method for linear elasticity with non-homogeneous Dirichlet conditions, Electron. Trans. Numer. Anal., 26 (2007), 421-438.
doi: 10.1080/00207177708922320.
|
[28]
|
G. N. Gatica, A Simple Introduction to the Mixed Finite Element Method: Theory and Applications, Springer Briefs in Mathematics, Springer, Cham, 2014.
doi: 10.1007/978-3-319-03695-3.
|
[29]
|
G. N. Gatica, R. Oyarzúa and F.-J. Sayas, Analysis of fully-mixed finite element methods for the Stokes-Darcy coupled problem, Math. Comp., 80 (2011), 1911-1948.
doi: 10.1090/S0025-5718-2011-02466-X.
|
[30]
|
P. Z. Huang, W. Q. Li and Z. Y. Si, Several iterative schemes for the stationary natural convection equations at different Rayleigh numbers, Numer. Methods Partial Differential Equations, 31 (2015), 761-776.
doi: 10.1002/num.21915.
|
[31]
|
K. Julien, S. Legg, J. McWilliams and J. Werne, Rapidly rotating turbulent Rayleigh-Bénard convection, J. Fluid Mech., 322 (1996), 243-273.
doi: 10.1017/S0022112096002789.
|
[32]
|
M. Kaddiri, M. Naïmi, A. Raji and M. Hasnaoui, Rayleigh-Bénard convection of non-Newtonian power-law fluids with temperature-dependent viscosity, Int. Schol. Res. Netw., (2012), 614712.
doi: 10.5402/2012/614712.
|
[33]
|
P. Mora and D. A. Yuen, Comparison of convection for Reynolds and Arrhenius temperature dependent viscosities, Fluid Mech. Res. Int., 2 (2018), 99-104.
doi: 10.15406/fmrij.2018.02.00025.
|
[34]
|
R. Oyarzúa and P. Zúñiga, Analysis of a conforming finite element method for the Boussinesq problem with temperature-dependent parameters, J. Comput. Appl. Math., 323 (2017), 71-94.
doi: 10.1016/j.cam.2017.04.009.
|
[35]
|
C. E. Pérez, J.-M. Thomas, S. Blancher and R. Creff, The steady Navier-Stokes/energy system with temperature-dependent viscosity. I: Analysis of the continuous problem, Internat. J. Numer. Methods Fluids, 56 (2008), 63-89.
doi: 10.1002/fld.1509.
|
[36]
|
C. E. Pérez, J.-M. Thomas, S. Blancher and R. Creff, The steady Navier-Stokes/energy system with temperature-dependent viscosity. II: The discrete problem and numerical experiments, Internat. J. Numer. Methods Fluids, 56 (2008), 91-114.
doi: 10.1002/fld.1572.
|
[37]
|
A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations, Springer Series in Computational Mathematics, 23. Springer-Verlag, Berlin, 1994.
|
[38]
|
J. E. Roberts and J. M. Thomas, Mixed and hybrid methods, Handbook of Numerical Analysis, Handb. Numer. Anal., II, North-Holland, Amsterdam, 2 (1991), 523-639.
|
[39]
|
M. Tabata and D. Tagami, Error estimates of finite element methods for nonstationary thermal convection problems with temperature-dependent coefficients, Numer. Math., 100 (2005), 351-372.
doi: 10.1007/s00211-005-0589-2.
|
[40]
|
C. Waluga, B. Wohlmuth and U. Rüde, Mass-corrections for the conservative coupling of flow and transport on collocated meshes, J. Comput. Phys., 305 (2016), 319-332.
doi: 10.1016/j.jcp.2015.10.044.
|
[41]
|
J. Woodfield, M. Alvarez, B. Gómez-Vargas and R. Ruiz-Baier, Stability and finite element approximation of phase change models for natural convection in porous media, J. Comput. Appl. Math., 360 (2019), 117-137.
doi: 10.1016/j.cam.2019.04.003.
|
[42]
|
T. Zhang and H. X. Liang, Decoupled stabilized finite element methods for the Boussinesq equations with temperature-dependent coefficients, Internat. J. Heat Mass Tr., 110 (2017), 151-165.
doi: 10.1016/j.ijheatmasstransfer.2017.03.002.
|
[43]
|
A. G. Zimmerman and J. Kowalski, Simulating convection-coupled phase-change in enthalpy form with mixed finite elements, Preprint, (2019), arXiv: 1907.0441v1.
|