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.
Mots clés : 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 }, pages = {1765--1781}, publisher = {EDP-Sciences}, volume = {47}, number = {6}, year = {2013}, doi = {10.1051/m2an/2013087}, mrnumber = {3123375}, zbl = {1293.65129}, language = {en}, url = {http://www.numdam.org/articles/10.1051/m2an/2013087/} }
TY - JOUR AU - Flotron, Stéphane AU - Rappaz, Jacques TI - Conservation schemes for convection-diffusion equations with Robin boundary conditions JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2013 SP - 1765 EP - 1781 VL - 47 IS - 6 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2013087/ DO - 10.1051/m2an/2013087 LA - en ID - M2AN_2013__47_6_1765_0 ER -
%0 Journal Article %A Flotron, Stéphane %A Rappaz, Jacques %T Conservation schemes for convection-diffusion equations with Robin boundary conditions %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2013 %P 1765-1781 %V 47 %N 6 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2013087/ %R 10.1051/m2an/2013087 %G en %F 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 , Tome 47 (2013) no. 6, pp. 1765-1781. doi : 10.1051/m2an/2013087. http://www.numdam.org/articles/10.1051/m2an/2013087/
[1] 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
, , and ,[2] Eigenvalue problems, Handbook of Numerical Analysis, vol. 2. Elsevier (1991) 641-787. | MR | Zbl
and ,[3] 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
and ,[4] Edge stabilization for Galerkin approximations of convection-diffusion-reaction problems. Comput. Methods Appl. Mech. Engrg. 193 (2004) 1437-1453 | MR | Zbl
and ,[5] The finite element method for elliptic problems. North-Holland Publishing Company (1978). | MR | Zbl
,[6] Chap XVIII. Evolution Problems: Variational Methods, Math. Anal. and Numer. Methods Sci. Technology. vol. 5, Springer-Verlag, Heidelberg (2000) 467-680.
and ,[7] Elements finis: Théorie, applications, mise en oeuvre. Springer-Verlag (2002). | MR | Zbl
and ,[8] Simulations numériques de phénomènes MHD-thermique avec interface libre dans l'électrolyse de l'aluminium, Ph.D. Thesis, EPFL, Switzerland, expected in (2013).
,[9] Numerical Simulation and optimization of the alumina distribution in an aluminium electrolysis pot, Ph.D. Thesis, Thesis No. 5023, EPFL, Switzerland (2011).
,[10] Numerical approximation of partial differential equations. Springer Series in Computational Mathematics (1997). | MR | Zbl
and ,[11] Navier-Stokes equations. North-Holland (1984). | MR | Zbl
,[12] Galerkin Finite Element Methods for Parabolic Problems. Springer Series in Computational Mathematics. Springer-Verlag Berlin Heidelberg, New York (1997). | MR | Zbl
,Cité par Sources :