main-content

Erschienen in:

01.12.2020

# A new mixed-FEM for steady-state natural convection models allowing conservation of momentum and thermal energy

verfasst von: Sergio Caucao, Ricardo Oyarzúa, Segundo Villa-Fuentes

Erschienen in: Calcolo | Ausgabe 4/2020

Einloggen, um Zugang zu erhalten

## Abstract

In this work we present a new mixed finite element method for a class of steady-state natural convection models describing the behavior of non-isothermal incompressible fluids subject to a heat source. Our approach is based on the introduction of a modified pseudostress tensor depending on the pressure, and the diffusive and convective terms of the Navier–Stokes equations for the fluid and a vector unknown involving the temperature, its gradient and the velocity. The introduction of these further unknowns lead to a mixed formulation where the aforementioned pseudostress tensor and vector unknown, together with the velocity and the temperature, are the main unknowns of the system. Then the associated Galerkin scheme can be defined by employing Raviart–Thomas elements of degree k for the pseudostress tensor and the vector unknown, and discontinuous piece-wise polynomial elements of degree k for the velocity and temperature. With this choice of spaces, both, momentum and thermal energy, are conserved if the external forces belong to the velocity and temperature discrete spaces, respectively, which constitutes one of the main feature of our approach. We prove unique solvability for both, the continuous and discrete problems and provide the corresponding convergence analysis. Further variables of interest, such as the fluid pressure, the fluid vorticity, the fluid velocity gradient, and the heat-flux can be easily approximated as a simple postprocess of the finite element solutions with the same rate of convergence. Finally, several numerical results illustrating the performance of the method are provided.
Literatur
1.
Allendes, A., Barrenechea, G.R., Narranjo, C.: A divergence-free low-order stabilized finite element method for a generalized steady state Boussinesq problem. Comput. Methods Appl. Mech. Eng. 340, 90–120 (2018)
2.
Almonacid, J.A., Gatica, G.N.: A fully-mixed finite element method for the n-dimensional Boussinesq problem with temperature-dependent parameters. Comput. Methods Appl. Math. 20(2), 187–213 (2020)
3.
Alvarez, M., Gatica, G.N., Ruiz-Baier, R.: An augmented mixed-primal finite element method for a coupled flow-transport problem. ESAIM Math. Model. Numer. Anal. 49(5), 1399–1427 (2015)
4.
Alvarez, M., Gatica, G.N., Ruiz-Baier, R.: A posteriori error analysis for a viscous flow-transport problem. ESAIM Math. Model. Numer. Anal. 50(6), 1789–1816 (2016)
5.
Barakos, G., Mitsoulis, E., Assimacopoulos, D.: Natural convection flow in a square cavity revisited: laminar and turbulent models with wall functions. Int. J. Numer. Methods Fluids 18, 695–719 (1994) CrossRef
6.
Bernardi, C., Métivet, B., Pernaud-Thomas, B.: Couplage des équations de Navier-Stokes et de la chaleur: le modèle et son approximation par éléments finis. (French) [Coupling of Navier-Stokes and heat equations: the model and its finite-element approximation] RAIRO Modél. Math. Anal. Numér. 29(7), 871–921 (1995)
7.
Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer Series in Computational Mathematics. Springer, New York (1991) CrossRef
8.
Bürger, R., Méndez, P.E., Ruiz-Baier, R.: On $$H(div)$$-conforming methods for double-diffusion equations in porous media. SIAM J. Numer. Anal. 57(3), 1318–1343 (2019)
9.
Camaño, J., García, C., Oyarzúa, R.: Analysis of a conservative mixed-FEM for the stationary Navier–Stokes problem. Submitted. [preliminary version available as Preprint 2018-25 in http://​www.​ci2ma.​udec.​cl/​publicaciones/​prepublicaciones​/​en.​php]
10.
Camaño, J., Muñoz, C., Oyarzúa, R.: Numerical analysis of a dual-mixed problem in non-standard Banach spaces. Electron. Trans. Numer. Anal. 48, 114–130 (2018)
11.
Camaño, J., Oyarzúa, R., Tierra, G.: Analysis of an augmented mixed-FEM for the Navier–Stokes problem. Math. Comput. 86(304), 589–615 (2017)
12.
Çibik, A., Kaya, S.: A projection-based stabilized finite element method for steady-state natural convection problem. J. Math. Anal. Appl. 381(2), 469–484 (2011)
13.
Cockburn, B., Kanschat, G., Schötzau, D.: A locally conservative LDG method for the incompressible Navier–Stokes equations. Math. Comput. 74(251), 1067–1095 (2005)
14.
Cockburn, B., Kanschat, G., Schötzau, D.: A note on discontinuous Galerkin divergence-free solutions of the Navier–Stokes equations. J. Sci. Comput. 31(1–2), 61–73 (2007)
15.
Colmenares, E., Gatica, G.N., Moraga, S.: A Banach spaces-based analysis of a new fully-mixed finite element method for the Boussinesq problem. ESAIM Math. Model. Numer. Anal. 54(5), 1525–1568 (2020)
16.
Colmenares, E., Gatica, G.N., Oyarzúa, R.: Analysis of an augmented mixed-primal formulation for the stationary Boussinesq problem. Numer. Methods Partial Differ. Equ. 32(2), 445–478 (2016)
17.
Colmenares, E., Gatica, G.N., Oyarzúa, R.: Fixed point strategies for mixed variational formulations of the stationary Boussinesq problem. C. R. Math. Acad. Sci. Paris 354(1), 57–62 (2016)
18.
Colmenares, E., Gatica, G.N., Oyarzúa, R.: An augmented fully-mixed finite element method for the stationary Boussinesq problem. Calcolo 54(1), 167–205 (2017)
19.
Colmenares, E., Neilan, M.: Dual-mixed finite element methods for the stationary Boussinesq problem. Comput. Math. Appl. 72(7), 1828–1850 (2016)
20.
Dalal, A., Das, M.K.: Natural convection in a rectangular cavity heated from below and uniformly cooled from the top and both sides. Numer. Heat Tr. A-Appl 49(3), 301–322 (2006) CrossRef
21.
Dallmann, H., Arndt, D.: Stabilized finite element methods for the Oberbeck–Boussinesq model. J. Sci. Comput. 69(1), 244–273 (2016)
22.
De Vahl Davis, G.: Natural convection of air in a square cavity: a bench mark numerical solution. Int. J. Numer. Methods Fluids 3, 249–264 (1983) CrossRef
23.
Ern, A., Guermond, J.-L.: Theory and Practice of Finite Elements. Applied Mathematical Sciences, 159. Springer, New York (2004)
24.
Farhoul, M., Nicaise, S., Paquet, L.: A mixed formulation of Boussinesq equations: analysis of nonsingular solutions. Math. Comput. 69(231), 965–986 (2000)
25.
Gatica, G.N.: A Simple Introduction to the Mixed Finite Element Method. Theory and Applications. SpringerBriefs in Mathematics, Springer, Cham (2014)
26.
Jakab, T., Mitrea, I., Mitrea, M.: Sobolev estimates for the Green potential associated with the Robin-Laplacian in Lipschitz domains satisfying a uniform exteriour ball condition, Sobolev Spaces in mathematics II, Applications in Analysis and Partial Differential Equations. International Mathematical Series, Vol. 9. Springer, Novosibirsk (2008)
27.
Oyarzúa, R., Qin, T., Schötzau, D.: An exactly divergence-free finite element method for a generalized Boussinesq problem. IMA J. Numer. Anal. 34(3), 1104–1135 (2014)
28.
Oyarzúa, R., Serón, M.: A divergence-conforming DG-mixed finite element method for the stationary Boussinesq problem. J. Sci. Comput. 85(1), 14 (2020)
29.
Oyarzúa, R., Zúñiga, P.: Analysis of a conforming finite element method for the Boussinesq problem with temperature-dependent parameters. J. Comput. Appl. Math. 323, 71–94 (2017)
30.
Pérez, C.E., Thomas, J.-M., Blancher, S., Creff, R.: The steady Navier–Stokes/energy system with temperature-dependent viscosity–Part 2: the discrete problem and numerical experiments. Int. J. Numer. Methods Fluids 56(1), 91–114 (2008) CrossRef
31.
Quarteroni, A., Valli, A.: Numerical Approximation of Partial Differential Equations. Springer Series in Computational Mathematics, 23. Springer, Berlin (1994)
32.
Tabata, M., Tagami, D.: Error estimates of finite element methods for nonstationary thermal convection problems with temperature-dependent coefficients. Numer. Math. 100(2), 351–372 (2005)
33.
Zhang, T., Liang, H.: Decoupled stabilized finite element methods for the Boussinesq equations with temperature-dependent coefficients. Int. J. Heat Mass Tr. 110, 151–165 (2017) CrossRef
Titel
A new mixed-FEM for steady-state natural convection models allowing conservation of momentum and thermal energy
verfasst von
Sergio Caucao
Ricardo Oyarzúa
Segundo Villa-Fuentes
Publikationsdatum
01.12.2020
Verlag
Springer International Publishing
Erschienen in
Calcolo / Ausgabe 4/2020
Print ISSN: 0008-0624
Elektronische ISSN: 1126-5434
DOI
https://doi.org/10.1007/s10092-020-00385-3

Zur Ausgabe