Conservation schemes for convection-diffusion equations with Robin boundary conditions
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 47 (2013) no. 6, pp. 1765-1781.

In this article, we present a numerical scheme based on a finite element method in order to solve a time-dependent convection-diffusion equation problem and satisfy some conservation properties. In particular, our scheme is able to conserve the total energy for a heat equation or the total mass of a solute in a fluid for a concentration equation, even if the approximation of the velocity field is not completely divergence-free. We establish a priori errror estimates for this scheme and we give some numerical examples which show the efficiency of the method.

DOI : 10.1051/m2an/2013087
Classification : 65M60, 35K20, 80A20
Mots clés : finite elements, numerical conservation schemes, Robin boundary condition, convection-diffusion equations
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     title = {Conservation schemes for convection-diffusion equations with {Robin} boundary conditions},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {1765--1781},
     publisher = {EDP-Sciences},
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Flotron, Stéphane; Rappaz, Jacques. Conservation schemes for convection-diffusion equations with Robin boundary conditions. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 47 (2013) no. 6, pp. 1765-1781. doi : 10.1051/m2an/2013087. http://www.numdam.org/articles/10.1051/m2an/2013087/

[1] P. Angot, V. Dolej, M. Feistauer and J. Felcman, Analysis of a combined barycentric finite volumenonconforming finite element method for nonlinear convection-diffusion problems, Applications of Mathematics, vol. 43. Kluwer Academic Publishers-Plenum Publishers (1998) 263-310. | MR | Zbl

[2] I. Babuska and J. Osborn, Eigenvalue problems, Handbook of Numerical Analysis, vol. 2. Elsevier (1991) 641-787. | MR | Zbl

[3] A. Brooks and T. Hughes, Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations. Comput. Methods Appl. Mech. Engrg. 32 (1982) 199-259 | MR | Zbl

[4] E. Burman and P. Hansbo, Edge stabilization for Galerkin approximations of convection-diffusion-reaction problems. Comput. Methods Appl. Mech. Engrg. 193 (2004) 1437-1453 | MR | Zbl

[5] P.G. Ciarlet, The finite element method for elliptic problems. North-Holland Publishing Company (1978). | MR | Zbl

[6] R. Dautray and J.-L. Lions, Chap XVIII. Evolution Problems: Variational Methods, Math. Anal. and Numer. Methods Sci. Technology. vol. 5, Springer-Verlag, Heidelberg (2000) 467-680.

[7] A. Ern and J.-L. Guermond, Elements finis: Théorie, applications, mise en oeuvre. Springer-Verlag (2002). | MR | Zbl

[8] S. Flotron, Simulations numériques de phénomènes MHD-thermique avec interface libre dans l'électrolyse de l'aluminium, Ph.D. Thesis, EPFL, Switzerland, expected in (2013).

[9] T. Hofer, Numerical Simulation and optimization of the alumina distribution in an aluminium electrolysis pot, Ph.D. Thesis, Thesis No. 5023, EPFL, Switzerland (2011).

[10] A. Quarteroni and A. Valli, Numerical approximation of partial differential equations. Springer Series in Computational Mathematics (1997). | MR | Zbl

[11] R. Temam, Navier-Stokes equations. North-Holland (1984). | MR | Zbl

[12] V. Thomée, Galerkin Finite Element Methods for Parabolic Problems. Springer Series in Computational Mathematics. Springer-Verlag Berlin Heidelberg, New York (1997). | MR | Zbl

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