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Conservation schemes for convection-diffusion equations withRobin boundary conditions∗∗∗

Published online by Cambridge University Press:  11 October 2013

Stéphane Flotron  and
Jacques Rappaz
Stéphane Flotron
Affiliation:
Ecole Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland.. stephane.flotron@epfl.ch; jacques.rappaz@epfl.ch
Jacques Rappaz
Affiliation:
Ecole Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland.. stephane.flotron@epfl.ch; jacques.rappaz@epfl.ch
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Abstract

In this article, we present a numerical scheme based on a finite element method in orderto solve a time-dependent convection-diffusion equation problem and satisfy someconservation properties. In particular, our scheme is able to conserve the total energyfor 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. Weestablish a priori errror estimates for this scheme and we give some numerical exampleswhich show the efficiency of the method.

Keywords

Finite Elementsnumerical conservation schemesRobin boundary conditionconvection-diffusion equations
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 47 , Issue 6 , November 2013 , pp. 1765 - 1781
Copyright
© EDP Sciences, SMAI 2013

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References

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.
I. Babuska and J. Osborn, Eigenvalue problems, Handbook of Numerical Analysis, vol. 2. Elsevier (1991) 641–787.
Brooks, A. and Hughes, T., 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 Google Scholar
Burman, E. and Hansbo, P., Edge stabilization for Galerkin approximations of convection–diffusion–reaction problems. Comput. Methods Appl. Mech. Engrg. 193 (2004) 1437–1453 Google Scholar
P.G. Ciarlet, The finite element method for elliptic problems. North-Holland Publishing Company (1978).
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.
A. Ern and J.-L. Guermond, Elements finis: Théorie, applications, mise en oeuvre. Springer-Verlag (2002).
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).
T. Hofer, Numerical Simulation and optimization of the alumina distribution in an aluminium electrolysis pot, Ph.D. Thesis, Thesis No. 5023, EPFL, Switzerland (2011).
A. Quarteroni and A. Valli, Numerical approximation of partial differential equations. Springer Series in Computational Mathematics (1997).
R. Temam, Navier-Stokes equations. North-Holland (1984).
V. Thomée, Galerkin Finite Element Methods for Parabolic Problems. Springer Series in Computational Mathematics. Springer-Verlag Berlin Heidelberg, New York (1997).