Spectral discretization of the Navier–Stokes equations coupled with the heat equation
Agroum, Rahma1 ; Aouadi, Saloua Mani2 ; Bernardi, Christine3 ; Satouri, Jamil4
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.
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. 
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     title = {Spectral discretization of the {Navier{\textendash}Stokes} equations coupled with the heat equation},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
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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://www.numdam.org/articles/10.1051/m2an/2014049/

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