A semidiscretization in time of a fourth order nonlinear parabolic system in several space dimensions arising in quantum semiconductor modelling is studied. The system is numerically treated by introducing an additional nonlinear potential. Exploiting the stability of the discretization, convergence is shown in the multi-dimensional case. Under some assumptions on the regularity of the solution, the rate of convergence proves to be optimal.
Mots-clés : higher order parabolic PDE, positivity, semidiscretization, stability, convergence, semiconductors
@article{M2AN_2003__37_2_277_0,
author = {J\"ungel, Ansgar and Pinnau, Ren\'e},
title = {Convergent semidiscretization of a nonlinear fourth order parabolic system},
journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
pages = {277--289},
publisher = {EDP-Sciences},
volume = {37},
number = {2},
year = {2003},
doi = {10.1051/m2an:2003026},
mrnumber = {1991201},
zbl = {1026.35045},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an:2003026/}
}
TY - JOUR AU - Jüngel, Ansgar AU - Pinnau, René TI - Convergent semidiscretization of a nonlinear fourth order parabolic system JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2003 SP - 277 EP - 289 VL - 37 IS - 2 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an:2003026/ DO - 10.1051/m2an:2003026 LA - en ID - M2AN_2003__37_2_277_0 ER -
%0 Journal Article %A Jüngel, Ansgar %A Pinnau, René %T Convergent semidiscretization of a nonlinear fourth order parabolic system %J ESAIM: Modélisation mathématique et analyse numérique %D 2003 %P 277-289 %V 37 %N 2 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an:2003026/ %R 10.1051/m2an:2003026 %G en %F M2AN_2003__37_2_277_0
Jüngel, Ansgar; Pinnau, René. Convergent semidiscretization of a nonlinear fourth order parabolic system. ESAIM: Modélisation mathématique et analyse numérique, Tome 37 (2003) no. 2, pp. 277-289. doi : 10.1051/m2an:2003026. http://www.numdam.org/articles/10.1051/m2an:2003026/
[1] , Sobolev Spaces. First edition, Academic Press, New York (1975). | MR | Zbl
[2] , Diffusion-drift modelling of strong inversion layers. COMPEL 6 (1987) 11-18.
[3] , and, Finite element approximation of a fourth order nonlinear degenerate parabolic equation. Numer. Math. 80 (1998) 525-556. | Zbl
[4] and, On the stationary quantum drift diffusion model. Z. Angew. Math. Phys. 49 (1998) 251-275. | Zbl
[5] and, Higher order nonlinear degenerate parabolic equations. J. Differential Equations 83 (1990) 179-206. | Zbl
[6] , The mathematics of moving contact lines in thin liquid films. Notices Amer. Math. Soc. 45 (1998) 689-697. | Zbl
[7] and, Long-wave instabilities and saturation in thin film equations. Comm. Pure Appl. Math. 51 (1998) 625-661. | Zbl
[8] and, Positivity preserving numerical schemes for lubriaction-typeequations. SIAM J. Numer. Anal. 37 (2000) 523-555. | Zbl
[9] , and, Existence and positivity of solutions of a fourth-order nonlinear PDE describing interface fluctuations. Comm. Pure Appl. Math. 47 (1994) 923-942. | Zbl
[10] and, Modular alorithms for transient semiconductor device simulation, part I: Analysis of the outer iteration, in Computational Aspects of VLSI Design with an Emphasis on Semiconductor Device Simulations, R.E. Bank Ed. (1990) 107-149. | Zbl
[11] , and, On a fourth-order degenerate parabolic equation: Global entropy estimates, existence and quantitative behavior of solutions. SIAM J. Math. Anal. 29 (1998) 321-342. | Zbl
[12] , The quantum hydrodynamic model for semiconductor devices. SIAM J. Appl. Math. 54 (1994) 409-427. | Zbl
[13] and, Approximation of thermal equilibrium for quantum gases with discontinuous potentials and applications to semiconductor devices. SIAM J. Appl. Math. 58 (1998) 780-805. | Zbl
[14] and, The quantum hydrodynamic model for semiconductors in thermal equilibrium. Z. Angew. Math. Phys. 48 (1997) 45-59. | Zbl
[15] and, Quantum hydrodynamics, Wigner transform and the classical limit. Asymptot. Anal. 14 (1997) 97-116. | Zbl
[16] and, Nonnegativity preserving convergent schemes for the thin film equation. Numer. Math. 87 (2000) 113-152. | Zbl
[17] and, A quantum regularization of the one-dimensional hydrodynamic model for semiconductors. Adv. Differential Equations 5 (2000) 773-800.
[18] , Quasi-hydrodynamic Semiconductor Equations. Birkhäuser, PNLDE 41 (2001). | MR | Zbl
[19] and, Global non-negative solutions of a nonlinear fourth order parabolic equation for quantum systems. SIAM J. Math. Anal. 32 (2000) 760-777. | Zbl
[20] and, A positivity preserving numerical scheme for a nonlinear fourth-order parabolic system. SIAM J. Numer. Anal. 39 (2001) 385-406. | Zbl
[21] , and, Semiconductor Equations. First edition, Springer-Verlag, Wien (1990). | Zbl
[22] and, A variational analysis of the thermal equilibrium state of charged quantum fluids. Comm. Partial Differential Equations 20 (1995) 885-900. | Zbl
[23] and, Weak limits of the quantum hydrodynamic model. To appear in Proc. International Workshop on Quantum Kinetic Theory.
[24] , A note on boundary conditions for quantum hydrodynamic models. Appl. Math. Lett. 12 (1999) 77-82. | Zbl
[25] , The linearized transient quantum drift diffusion model - stability of stationary states. ZAMM 80 (2000) 327-344. | Zbl
[26] , Numerical study of the Quantum Euler-Poisson model. To appear in Appl. Math. Lett. | Zbl
[27] and, The stationary current-voltage characteristics of the quantum drift diffusion model. SIAM J. Numer. Anal. 37 (1999) 211-245. | Zbl
[28] , Compact sets in the space . Ann. Mat. Pura Appl. 146 (1987) 65-96. | Zbl
[29] , Elliptic Differential Equations and Obstacle Problems. First edition, Plenum Press, New York (1987). | MR | Zbl
Cité par Sources :
