Spectral discretization of the Navier–Stokes equations coupled with the heat equation
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 3, pp. 621-639.

We consider the spectral discretization of the Navier–Stokes equations coupled with the heat equation where the viscosity depends on the temperature, with boundary conditions which involve the velocity and the temperature. This problem admits a variational formulation with three independent unknowns, the velocity, the pressure and the temperature. We prove optimal error estimates and present some numerical experiments which confirm the validity of the discretization.

Reçu le :
DOI : 10.1051/m2an/2014049
Classification : 35K05, 36Q30, 80M22
Mots-clés : Navier–Stokes equations, heat equation, spectral methods
Agroum, Rahma 1 ; Aouadi, Saloua Mani 2 ; Bernardi, Christine 3 ; Satouri, Jamil 4

1 Faculty of Sciences of Tunis, University of Tunis El Manar, 2060 Tunis, Tunisia.Presently at Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, 4 place Jussieu, 75252 Paris cedex 05, France.
2 Faculty of Sciences of Tunis, University of Tunis El Manar, 2060 Tunis, Tunisia.
3 Laboratoire Jacques-Louis Lions, C.N.R.S and Université Pierre et Marie Curie, Boîte courrier 187, 4 place Jussieu, 75252 Paris cedex 05, France.
4 IPEIT-University of Tunis, 2 street Jawaher Lel Nehru-1089, Monfleury, Tunisia.
@article{M2AN_2015__49_3_621_0,
     author = {Agroum, Rahma and Aouadi, Saloua Mani and Bernardi, Christine and Satouri, Jamil},
     title = {Spectral discretization of the {Navier{\textendash}Stokes} equations coupled with the heat equation},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {621--639},
     publisher = {EDP-Sciences},
     volume = {49},
     number = {3},
     year = {2015},
     doi = {10.1051/m2an/2014049},
     zbl = {1325.35134},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/m2an/2014049/}
}
TY  - JOUR
AU  - Agroum, Rahma
AU  - Aouadi, Saloua Mani
AU  - Bernardi, Christine
AU  - Satouri, Jamil
TI  - Spectral discretization of the Navier–Stokes equations coupled with the heat equation
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2015
SP  - 621
EP  - 639
VL  - 49
IS  - 3
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/m2an/2014049/
DO  - 10.1051/m2an/2014049
LA  - en
ID  - M2AN_2015__49_3_621_0
ER  - 
%0 Journal Article
%A Agroum, Rahma
%A Aouadi, Saloua Mani
%A Bernardi, Christine
%A Satouri, Jamil
%T Spectral discretization of the Navier–Stokes equations coupled with the heat equation
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2015
%P 621-639
%V 49
%N 3
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/m2an/2014049/
%R 10.1051/m2an/2014049
%G en
%F M2AN_2015__49_3_621_0
Agroum, Rahma; Aouadi, Saloua Mani; Bernardi, Christine; Satouri, Jamil. Spectral discretization of the Navier–Stokes equations coupled with the heat equation. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 3, pp. 621-639. doi : 10.1051/m2an/2014049. http://archive.numdam.org/articles/10.1051/m2an/2014049/

C. Bernardi and Y. Maday, Spectral Methods, in the Handb. Numer. Anal. V. Edited by P.G. Ciarlet and J.-L. Lions. North-Holland (1997) 209–485.

C. Bernardi, Y, Maday and F. Rapetti, Discrétisations variationnelles de problèmes aux limites elliptiques. In vol. 45 of Collection Math. Appl. Springer-Verlag (2004). | Zbl

C. Bernardi, F. Coquel and P.-A. Raviart, Automatic coupling and finite element discretization of the Navier–Stokes and heat equations, Internal Report 10001, Laboratoire Jacques-Louis Lions (2010).

F. Brezzi, J. Rappaz and P.-A. Raviart, Finite dimensionnal approximation of nonlinear problems, Part I: Branches of nonsingular solutions. Numer. Math. 36 (1980) 1–25. | DOI | Zbl

F. Brezzi, C. Canuto and A. Russo, A self-adaptive formulation for the Euler/Navier–Stokes coupling. Comput. Methods Appl. Mech. Engrg. 73 (1989) 317–330. | DOI | Zbl

M. Bulíček, E. Feireisl and J. Málek, A Navier–Stokes-Fourier system for incompressible fluids with temperature dependent material coefficients. Nonlin. Anal. Real World Appl. 10 (2009) 992–1015. | DOI | Zbl

M. Dauge, Problèmes de Neumann et de Dirichlet sur un polyèdre dans R 3 : régularité dans les espaces de Sobolev L p . C.R. Acad. Sci. Paris Sér. I 307 (1988) 27–32. | Zbl

M. Dauge, Neumann and mixed problems on curvilinear polyhedra. Integral Equations Oper. Theory 15 (1992) 227–261. | DOI | Zbl

V. Girault and P.-A. Raviart, Finite Element Methods for Navier–Stokes Equations. Theory and Algorithms. Springer Ser. Comput. Math. Springer-Verlag, Berlin (1986). | Zbl

P. Grisvard, Elliptic Problems in Nonsmooth Domains. Pitman (1985). | Zbl

Y. Maday and E.M. Rønquist, Optimal error analysis of spectral methods with emphasis on non-constant coefficients and deformed geometries. Comput. Methods Appl. Mech. Engrg. 80 (1990) 91–115. | DOI | Zbl

N.G. Meyers, An L p -estimate for the gradient of solutions of second order elliptic divergence equations. Ann. Sc. Norm. Sup. Pisa 17 (1963) 189–206. | Numdam | Zbl

A. Quarteroni, Some results of Bernstein and Jackson type for polynomial approximation in L p -spaces. Japan J. Appl. Math. 1 (1984) 173–181. | DOI | Zbl

G. Talenti, Best constant in Sobolev inequality. Ann. Math. Pura ed Appl. Ser. IV 110 (1976) 353–372. | DOI | Zbl

Cité par Sources :