Conservation schemes for convection-diffusion equations with Robin boundary conditions

Stéphane Flotron; Jacques Rappaz

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2013)

  • Volume: 47, Issue: 6, page 1765-1781
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Flotron, Stéphane, and Rappaz, Jacques. "Conservation schemes for convection-diffusion equations with Robin boundary conditions." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 47.6 (2013): 1765-1781. <http://eudml.org/doc/273326>.

@article{Flotron2013,
abstract = {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.},
author = {Flotron, Stéphane, Rappaz, Jacques},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {finite elements; numerical conservation schemes; Robin boundary condition; convection-diffusion equations; Robin boundary conditions; finite element method; error bounds; stability; numerical experiment},
language = {eng},
number = {6},
pages = {1765-1781},
publisher = {EDP-Sciences},
title = {Conservation schemes for convection-diffusion equations with Robin boundary conditions},
url = {http://eudml.org/doc/273326},
volume = {47},
year = {2013},
}

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 - Modélisation Mathématique et Analyse Numérique
PY - 2013
PB - EDP-Sciences
VL - 47
IS - 6
SP - 1765
EP - 1781
AB - 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.
LA - eng
KW - finite elements; numerical conservation schemes; Robin boundary condition; convection-diffusion equations; Robin boundary conditions; finite element method; error bounds; stability; numerical experiment
UR - http://eudml.org/doc/273326
ER -

References

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  4. [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 Zbl1085.76033MR2068903
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  7. [7] A. Ern and J.-L. Guermond, Elements finis: Théorie, applications, mise en oeuvre. Springer-Verlag (2002). Zbl0993.65123MR1933883
  8. [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. [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. [10] A. Quarteroni and A. Valli, Numerical approximation of partial differential equations. Springer Series in Computational Mathematics (1997). Zbl0803.65088MR1299729
  11. [11] R. Temam, Navier-Stokes equations. North-Holland (1984). Zbl0568.35002MR603444
  12. [12] V. Thomée, Galerkin Finite Element Methods for Parabolic Problems. Springer Series in Computational Mathematics. Springer-Verlag Berlin Heidelberg, New York (1997). Zbl0528.65052MR744045

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