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.
@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: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
volume = {37},
year = {2003},
pages = {277-289},
doi = {10.1051/m2an:2003026},
mrnumber = {1991201},
zbl = {1026.35045},
language = {en},
url = {http://dml.mathdoc.fr/item/M2AN_2003__37_2_277_0}
}
Jüngel, Ansgar; Pinnau, René. Convergent semidiscretization of a nonlinear fourth order parabolic system. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 37 (2003) pp. 277-289. doi : 10.1051/m2an:2003026. http://gdmltest.u-ga.fr/item/M2AN_2003__37_2_277_0/
[1] , Sobolev Spaces. First edition, Academic Press, New York (1975). | MR 450957 | Zbl 0314.46030
[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 0913.65084
[4] and, On the stationary quantum drift diffusion model. Z. Angew. Math. Phys. 49 (1998) 251-275. | Zbl 0936.35057
[5] and, Higher order nonlinear degenerate parabolic equations. J. Differential Equations 83 (1990) 179-206. | Zbl 0702.35143
[6] , The mathematics of moving contact lines in thin liquid films. Notices Amer. Math. Soc. 45 (1998) 689-697. | Zbl 0917.35100
[7] and, Long-wave instabilities and saturation in thin film equations. Comm. Pure Appl. Math. 51 (1998) 625-661. | Zbl 0916.35008
[8] and, Positivity preserving numerical schemes for lubriaction-typeequations. SIAM J. Numer. Anal. 37 (2000) 523-555. | Zbl 0961.76060
[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 0806.35059
[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 0692.65067
[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 0929.35061
[12] , The quantum hydrodynamic model for semiconductor devices. SIAM J. Appl. Math. 54 (1994) 409-427. | Zbl 0815.35111
[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 0957.76099
[14] and, The quantum hydrodynamic model for semiconductors in thermal equilibrium. Z. Angew. Math. Phys. 48 (1997) 45-59. | Zbl 0882.76108
[15] and, Quantum hydrodynamics, Wigner transform and the classical limit. Asymptot. Anal. 14 (1997) 97-116. | Zbl 0877.76087
[16] and, Nonnegativity preserving convergent schemes for the thin film equation. Numer. Math. 87 (2000) 113-152. | Zbl 0988.76056
[17] and, A quantum regularization of the one-dimensional hydrodynamic model for semiconductors. Adv. Differential Equations 5 (2000) 773-800. | Zbl pre01700749
[18] , Quasi-hydrodynamic Semiconductor Equations. Birkhäuser, PNLDE 41 (2001). | MR 1818867 | Zbl 0969.35001
[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 0979.35061
[20] and, A positivity preserving numerical scheme for a nonlinear fourth-order parabolic system. SIAM J. Numer. Anal. 39 (2001) 385-406. | Zbl 0994.35047
[21] , and, Semiconductor Equations. First edition, Springer-Verlag, Wien (1990). | Zbl 0765.35001
[22] and, A variational analysis of the thermal equilibrium state of charged quantum fluids. Comm. Partial Differential Equations 20 (1995) 885-900. | Zbl 0820.35112
[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 0952.76100
[25] , The linearized transient quantum drift diffusion model - stability of stationary states. ZAMM 80 (2000) 327-344. | Zbl 0947.35166
[26] , Numerical study of the Quantum Euler-Poisson model. To appear in Appl. Math. Lett. | Zbl 1077.82022
[27] and, The stationary current-voltage characteristics of the quantum drift diffusion model. SIAM J. Numer. Anal. 37 (1999) 211-245. | Zbl 0981.65076
[28] , Compact sets in the space . Ann. Mat. Pura Appl. 146 (1987) 65-96. | Zbl 0629.46031
[29] , Elliptic Differential Equations and Obstacle Problems. First edition, Plenum Press, New York (1987). | MR 1094820 | Zbl 0655.35002