Finite volume scheme for multi-dimensional drift-diffusion equations and convergence analysis
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 37 (2003) no. 2, p. 319-338

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.

DOI : https://doi.org/10.1051/m2an:2003028
Classification:  65M60,  76X05
Keywords: finite volume scheme, drift-diffusion equations, approximation of gradient
@article{M2AN_2003__37_2_319_0,
author = {Chainais-Hillairet, Claire and Liu, Jian-Guo and Peng, Yue-Jun},
title = {Finite volume scheme for multi-dimensional drift-diffusion equations and convergence analysis},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {37},
number = {2},
year = {2003},
pages = {319-338},
doi = {10.1051/m2an:2003028},
zbl = {1032.82038},
mrnumber = {1991203},
language = {en},
url = {http://www.numdam.org/item/M2AN_2003__37_2_319_0}
}

Chainais-Hillairet, Claire; Liu, Jian-Guo; Peng, Yue-Jun. Finite volume scheme for multi-dimensional drift-diffusion equations and convergence analysis. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 37 (2003) no. 2, pp. 319-338. doi : 10.1051/m2an:2003028. http://www.numdam.org/item/M2AN_2003__37_2_319_0/

[0] 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).

[0] H. Beirão Da Veiga, On the semiconductor drift diffusion equations. Differential Integral Equations 9 (1996) 729-744. | Zbl 0859.35055

[0] H. Brezis, Analyse Fonctionnelle - Théorie et Applications. Masson, Paris (1983). | Zbl 0511.46001

[0] F. Brezzi, L.D. Marini and P. Pietra, Méthodes d'éléments finis mixtes et schéma de Scharfetter-Gummel. C. R. Acad. Sci. Paris Sér. I Math. 305 (1987) 599-604. | Zbl 0623.65131

[0] F. Brezzi, L.D. Marini and P. Pietra, Two-dimensional exponential fitting and applications to drift-diffusion models. SIAM J. Numer. Anal. 26 (1989) 1342-1355. | Zbl 0686.65088

[0] C. Chainais-Hillairet and Y.J. Peng, Convergence of a finite volume scheme for the drift-diffusion equations in 1-D. IMA J. Numer. Anal. 23 (2003) 81-108. | Zbl 1018.65109

[0] 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. | Zbl 1072.82574

[0] 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). | MR 2047580 | Zbl 1127.65319

[0] P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978). | MR 520174 | Zbl 0383.65058

[0] R. Eymard, T. Gallouët and R. Herbin, Finite Volume Methods. North-Holland, Amsterdam, Handb. Numer. Anal. VII (2000) 713-1020. | Zbl 0981.65095

[0] R. Eymard, T. Gallouët, R. Herbin and A. Michel, Convergence of a finite volume scheme for nonlinear degenerate parabolic equations. Numer. Math. 92 (2002) 41-82. | Zbl 1005.65099

[0] W. Fang and K. Ito, Global solutions of the time-dependent drift-diffusion semiconductor equations. J. Differential Equations 123 (1995) 523-566. | Zbl 0845.35050

[0] H. Gajewski, On the uniqueness of solutions to the drift-diffusion model of semiconductor devices. Math. Models Methods Appl. Sci. 4 (1994) 121-133. | Zbl 0801.35133

[0] A. Jüngel, Numerical approximation of a drift-diffusion model for semiconductors with nonlinear diffusion. ZAMM Z. Angew. Math. Mech. 75 (1995) 783-799. | Zbl 0866.35056

[0] A. Jüngel, A nonlinear drift-diffusion system with electric convection arising in semiconductor and electrophoretic modeling. Math. Nachr. 185 (1997) 85-110. | Zbl pre01019611

[0] A. Jüngel and Y.J. Peng, A hierarchy of hydrodynamic models for plasmas: zero-relaxation-time limits. Comm. Partial Differential Equations 24 (1999) 1007-1033. | Zbl 0946.35074

[0] A. Jüngel and Y.J. Peng, Zero-relaxation-time limits in the hydrodynamic equations for plasmas revisited. Z. Angew. Math. Phys. 51 (2000) 385-396. | Zbl 0963.35115

[0] A. Jüngel and P. Pietra, A discretization scheme for a quasi-hydrodynamic semiconductor model. Math. Models Methods Appl. Sci. 7 (1997) 935-955. | Zbl 0907.35075

[0] P.A. Markowich, C.A. Ringhofer and C. Schmeiser, Semiconductor Equations. Springer-Verlag, Vienna (1990). | MR 1063852 | Zbl 0765.35001