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Finite volume scheme for multi-dimensionaldrift-diffusion equations and convergence analysis

Published online by Cambridge University Press:  15 November 2003

Claire Chainais-Hillairet ,
Jian-Guo Liu  and
Yue-Jun Peng
Claire Chainais-Hillairet
Affiliation:
Laboratoire de Mathématiques Appliquées, CNRS UMR 6620, Université Blaise Pascal (Clermont-Ferrand 2), 63177 Aubière Cedex, France. chainais@math.univ-bpclermont.fr., peng@math.univ-bpclermont.fr.
Jian-Guo Liu
Affiliation:
Institute for Physical Science and Technology and Department of Mathematics, University of Maryland, College Park, Maryland 20742, USA. jliu@math.umd.edu.
Yue-Jun Peng
Affiliation:
Laboratoire de Mathématiques Appliquées, CNRS UMR 6620, Université Blaise Pascal (Clermont-Ferrand 2), 63177 Aubière Cedex, France. chainais@math.univ-bpclermont.fr., peng@math.univ-bpclermont.fr.
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Abstract

We introduce a finite volume scheme for multi-dimensional drift-diffusion equations. Such equations arise from the theory of semiconductors and are composed of two continuity equations coupled with a Poisson equation. In the case that the continuity equations are non degenerate, we prove the convergence of the scheme and then the existence of solutions to the problem. The key point of the proof relies on the construction of an approximate gradient of the electric potential which allows us to deal with coupled terms in the continuity equations. Finally, a numerical example is given to show the efficiency of the scheme.

Keywords

Finite volume schemedrift-diffusion equationsapproximation of gradient.

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Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 37 , Issue 2 , March 2003 , pp. 319 - 338
Copyright
© EDP Sciences, SMAI, 2003

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References

F. Arimburgo, C. Baiocchi and L.D. Marini, Numerical approximation of the 1-D nonlinear drift-diffusion model in semiconductors, in Nonlinear kinetic theory and mathematical aspects of hyperbolic systems, Rapallo, (1992) 1-10. World Sci. Publishing, River Edge, NJ (1992).
Beir, H. ao da Veiga, On the semiconductor drift diffusion equations. Differential Integral Equations 9 (1996) 729-744.
H. Brezis, Analyse Fonctionnelle - Théorie et Applications. Masson, Paris (1983).
Brezzi, F., Marini, L.D. and Pietra, P., Méthodes d'éléments finis mixtes et schéma de Scharfetter-Gummel. C. R. Acad. Sci. Paris Sér. I Math. 305 (1987) 599-604.
Brezzi, F., Marini, L.D. and Pietra, P., Two-dimensional exponential fitting and applications to drift-diffusion models. SIAM J. Numer. Anal. 26 (1989) 1342-1355. CrossRef
Chainais-Hillairet, C. and Peng, Y.J., Convergence of a finite volume scheme for the drift-diffusion equations in 1-D. IMA J. Numer. Anal. 23 (2003) 81-108. CrossRef
C. Chainais-Hillairet and Y.J. Peng, A finite volume scheme to the drift-diffusion equations for semiconductors, in Proc. of The Third International Symposium on Finite Volumes for Complex Applications, R. Herbin and D. Kröner Eds., Hermes, Porquerolles, France (2002) 163-170.
C. Chainais-Hillairet and Y.J. Peng, Finite volume approximation for degenerate drift-diffusion system in several space dimensions. Math. Models Methods. Appl. Sci. (submitted).
P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978).
R. Eymard, T. Gallouët and R. Herbin, Finite Volume Methods. North-Holland, Amsterdam, Handb. Numer. Anal. VII (2000) 713-1020.
Eymard, R., Gallouët, T., Herbin, R. and Michel, A., Convergence of a finite volume scheme for nonlinear degenerate parabolic equations. Numer. Math. 92 (2002) 41-82. CrossRef
Fang, W. and Ito, K., Global solutions of the time-dependent drift-diffusion semiconductor equations. J. Differential Equations 123 (1995) 523-566. CrossRef
Gajewski, H., On the uniqueness of solutions to the drift-diffusion model of semiconductor devices. Math. Models Methods Appl. Sci. 4 (1994) 121-133. CrossRef
Jüngel, A., Numerical approximation of a drift-diffusion model for semiconductors with nonlinear diffusion. ZAMM Z. Angew. Math. Mech. 75 (1995) 783-799. CrossRef
Jüngel, A., A nonlinear drift-diffusion system with electric convection arising in semiconductor and electrophoretic modeling. Math. Nachr. 185 (1997) 85-110. CrossRef
Jüngel, A. and Peng, Y.J., A hierarchy of hydrodynamic models for plasmas: zero-relaxation-time limits. Comm. Partial Differential Equations 24 (1999) 1007-1033. CrossRef
Jüngel, A. and Peng, Y.J., Zero-relaxation-time limits in the hydrodynamic equations for plasmas revisited. Z. Angew. Math. Phys. 51 (2000) 385-396. CrossRef
Jüngel, A. and Pietra, P., A discretization scheme for a quasi-hydrodynamic semiconductor model. Math. Models Methods Appl. Sci. 7 (1997) 935-955. CrossRef
P.A. Markowich, C.A. Ringhofer and C. Schmeiser, Semiconductor Equations. Springer-Verlag, Vienna (1990).