Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-28T17:04:16.439Z Has data issue: false hasContentIssue false
  • English
  • Français

Convergent semidiscretization of a nonlinear fourth order parabolic system

Published online by Cambridge University Press:  15 November 2003

Ansgar Jüngel  and
René Pinnau
Ansgar Jüngel
Affiliation:
Fachbereich Mathematik, Universität Konstanz, 78457 Konstanz, Germany. juengel@mathematik.uni-konstanz.de.
René Pinnau
Affiliation:
Fachbereich Mathematik, Technische Universität Darmstadt, 64289 Darmstadt, Germany. pinnau@mathematik.tu-darmstadt.de.
Get access

Abstract

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.

Keywords

Higher order parabolic PDEpositivitysemidiscretizationstabilityconvergencesemiconductors.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 37 , Issue 2 , March 2003 , pp. 277 - 289
Copyright
© EDP Sciences, SMAI, 2003

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

R.A. Adams, Sobolev Spaces. First edition, Academic Press, New York (1975).
Ancona, M.G., Diffusion-drift modelling of strong inversion layers. COMPEL 6 (1987) 11-18. CrossRef
Barrett, J., Blowey, J. and Garcke, H., Finite element approximation of a fourth order nonlinear degenerate parabolic equation. Numer. Math. 80 (1998) 525-556. CrossRef
Ben Abdallah, N. and Unterreiter, A., On the stationary quantum drift diffusion model. Z. Angew. Math. Phys. 49 (1998) 251-275. CrossRef
Bernis, F. and Friedman, A., Higher order nonlinear degenerate parabolic equations. J. Differential Equations 83 (1990) 179-206. CrossRef
Bertozzi, A.L., The mathematics of moving contact lines in thin liquid films. Notices Amer. Math. Soc. 45 (1998) 689-697.
Bertozzi, A.L. and Pugh, M.C., Long-wave instabilities and saturation in thin film equations. Comm. Pure Appl. Math. 51 (1998) 625-661. 3.0.CO;2-9>CrossRef
Bertozzi, A.L. and Zhornitskaya, L., Positivity preserving numerical schemes for lubriaction-typeequations. SIAM J. Numer. Anal. 37 (2000) 523-555.
Bleher, P.M., Lebowitz, J.L. and Speer, E.R., Existence and positivity of solutions of a fourth-order nonlinear PDE describing interface fluctuations. Comm. Pure Appl. Math. 47 (1994) 923-942. CrossRef
W.M. Coughran and J.W. Jerome, 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.
Dal Passo, R., Garcke, H. and Grün, G., On a fourth-order degenerate parabolic equation: Global entropy estimates, existence and quantitative behavior of solutions. SIAM J. Math. Anal. 29 (1998) 321-342. CrossRef
Gardner, C.L., The quantum hydrodynamic model for semiconductor devices. SIAM J. Appl. Math. 54 (1994) 409-427. CrossRef
Gardner, C.L. and Ringhofer, Ch., Approximation of thermal equilibrium for quantum gases with discontinuous potentials and applications to semiconductor devices. SIAM J. Appl. Math. 58 (1998) 780-805. CrossRef
Gasser, I. and Jüngel, A., The quantum hydrodynamic model for semiconductors in thermal equilibrium. Z. Angew. Math. Phys. 48 (1997) 45-59. CrossRef
Gasser, I. and Markowich, P.A., Quantum hydrodynamics, Wigner transform and the classical limit. Asymptot. Anal. 14 (1997) 97-116.
Grün, G. and Rumpf, M., Nonnegativity preserving convergent schemes for the thin film equation. Numer. Math. 87 (2000) 113-152.
Gyi, M.T. and Jüngel, A., A quantum regularization of the one-dimensional hydrodynamic model for semiconductors. Adv. Differential Equations 5 (2000) 773-800.
A. Jüngel, Quasi-hydrodynamic Semiconductor Equations. Birkhäuser, PNLDE 41 (2001).
Jüngel, A. and Pinnau, R., Global non-negative solutions of a nonlinear fourth order parabolic equation for quantum systems. SIAM J. Math. Anal. 32 (2000) 760-777. CrossRef
Jüngel, A. and Pinnau, R., A positivity preserving numerical scheme for a nonlinear fourth-order parabolic system. SIAM J. Numer. Anal. 39 (2001) 385-406. CrossRef
P.A. Markowich, Ch. A. Ringhofer and Ch. Schmeiser, Semiconductor Equations. First edition, Springer-Verlag, Wien (1990).
Pacard, F. and Unterreiter, A., A variational analysis of the thermal equilibrium state of charged quantum fluids. Comm. Partial Differential Equations 20 (1995) 885-900. CrossRef
P. Pietra and C. Pohl, Weak limits of the quantum hydrodynamic model. To appear in Proc. International Workshop on Quantum Kinetic Theory.
Pinnau, R., A note on boundary conditions for quantum hydrodynamic models. Appl. Math. Lett. 12 (1999) 77-82. CrossRef
Pinnau, R., The linearized transient quantum drift diffusion model - stability of stationary states. ZAMM 80 (2000) 327-344. 3.0.CO;2-H>CrossRef
R. Pinnau, Numerical study of the Quantum Euler-Poisson model. To appear in Appl. Math. Lett.
Pinnau, R. and Unterreiter, A., The stationary current-voltage characteristics of the quantum drift diffusion model. SIAM J. Numer. Anal. 37 (1999) 211-245. CrossRef
J. Simon, Compact sets in the space L p (0,T;B). Ann. Mat. Pura Appl. 146 (1987) 65-96.
G.M. Troianiello, Elliptic Differential Equations and Obstacle Problems. First edition, Plenum Press, New York (1987).