Conservation schemes for convection-diffusion equations with Robin boundary conditions
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 47 (2013) no. 6, p. 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 : https://doi.org/10.1051/m2an/2013087
Classification:  65M60,  35K20,  80A20
Keywords: finite elements, numerical conservation schemes, Robin boundary condition, convection-diffusion equations
@article{M2AN_2013__47_6_1765_0,
     author = {Flotron, St\'ephane and Rappaz, Jacques},
     title = {Conservation schemes for convection-diffusion equations with Robin boundary conditions},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {47},
     number = {6},
     year = {2013},
     pages = {1765-1781},
     doi = {10.1051/m2an/2013087},
     zbl = {1293.65129},
     mrnumber = {3123375},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2013__47_6_1765_0}
}
Flotron, Stéphane; Rappaz, Jacques. Conservation schemes for convection-diffusion equations with Robin boundary conditions. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 47 (2013) no. 6, pp. 1765-1781. doi : 10.1051/m2an/2013087. http://www.numdam.org/item/M2AN_2013__47_6_1765_0/

[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 1627989 | Zbl 0942.76035

[2] I. Babuska and J. Osborn, Eigenvalue problems, Handbook of Numerical Analysis, vol. 2. Elsevier (1991) 641-787. | MR 1115240 | Zbl 0875.65087

[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 679322 | Zbl 0497.76041

[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 2068903 | Zbl 1085.76033

[5] P.G. Ciarlet, The finite element method for elliptic problems. North-Holland Publishing Company (1978). | MR 520174 | Zbl 0511.65078

[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 1933883 | Zbl 0993.65123

[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 1299729 | Zbl 0803.65088

[11] R. Temam, Navier-Stokes equations. North-Holland (1984). | MR 603444 | Zbl 0568.35002

[12] V. Thomée, Galerkin Finite Element Methods for Parabolic Problems. Springer Series in Computational Mathematics. Springer-Verlag Berlin Heidelberg, New York (1997). | MR 744045 | Zbl 0528.65052