Convergent semidiscretization of a nonlinear fourth order parabolic system

Mathematical Modelling and Numerical Analysis

Volumn 37, Issue 2, 2003, Pages 277-289

  Jüngel, Ansgar ^a   Pinnau, René ^b  

^a UNIVERSITY OF KONSTANZ   (Germany)

^b DARMSTADT UNIVERSITY OF TECHNOLOGY   (Germany)

Author keywords

Convergence; Higher order parabolic PDE; Positivity; Semiconductors; Semidiscretization; Stability

Indexed keywords

EID: 30244494628     PISSN: 0764583X     EISSN: None     Source Type: Journal    
DOI: 10.1051/m2an:2003026     Document Type: Article

References (29)

1. R.A. Adams, Sobolev Spaces. First edition, Academic Press, New York (1975).
2. M.G. Ancona, Diffusion-drift modelling of strong inversion layers. COMPEL 6 (1987) 11-18.
3. J. Barrett, J. Blowey and H. Garcke, Finite element approximation of a fourth order nonlinear degenerate parabolic equation.Numer. Math. 80 (1998) 525-556.
4. N. Ben Abdallah and A. Unterreiter, On the stationary quantum drift diffusion model. Z. Angew. Math, Phys. 49 (1998) 251-275.
5. F. Bernis and A. Friedman, Higher order nonlinear degenerate parabolic equations. J. Differential Equations 83 (1990) 179-206.
6. A.L. Bertozzi, The mathematics of moving contact lines in thin liquid films. Notices Amer. Math. Soc. 45 (1998) 689-697.
7. A.L. Bertozzi and M.C. Pugh, Long-wave instabilities and saturation in thin film equations. Comm. Pure Appl. Math. 51 (1998) 625-661.
8. A.L. Bertozzi and L. Zhornitskaya, Positivity preserving numerical schemes for lubriaction-typeequations. SIAM J. Numer. Anal. 37 (2000) 523-555.
9. P.M. Bleher, J.L. Lebowitz and E.R. Speer, Existence and positivity of solutions of a fourth-order nonlinear PDE describing interface fluctuations. Comm. Pure Appl. Math. 47 (1994) 923-942.
10. 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.
11. R. Dal Passo, H. Garcke and G. Grün, On a fourth-order degenerate parabolic equation: Global entropy estimates, existence and quantitative behavior of solutions. SIAM J. Math. Anal. 29 (1998) 321-342.
12. C.L. Gardner, The quantum hydrodynamic model for semiconductor devices. SIAM J. Appl. Math. 54 (1994) 409-427.
13. C.L. Gardner and Ch. Ringhofer, Approximation of thermal equilibrium for quantum gases with discontinuous potentials and applications to semiconductor devices. SIAM J. Appl. Math. 58 (1998) 780-805.
14. I. Gasser and A. Jüngel, The quantum hydrodynamic model for semiconductors in thermal equilibrium. Z. Angew. Math. Phys. 48 (1997) 45-59.
15. I. Gasser and P.A. Markowich, Quantum hydrodynamics, Wigner transform and the classical limit. Asymptot. Anal. 14 (1997) 97-116.
16. G. Grün and M. Rumpf, Nonnegativity preserving convergent schemes for the thin film equation. Numer. Math. 87 (2000) 113-152.
17. M.T. Gyi and A. Jüngel, A quantum regularization of the one-dimensional hydrodynamic model for semiconductors. Adv. Differential Equations 5 (2000) 773-800.
18. A. Jüngel, Quasi-hydrodynamic Semiconductor Equations. Birkhäuser, PNLDE 41 (2001).
19. A. Jüngel and R. Pinnau, Global non-negative solutions of a nonlinear fourth order parabolic equation for quantum systems. SIAM J. Math. Anal. 32 (2000) 760-777.
20. A. Jüngel and R. Pinnau, A positivity preserving numerical scheme for a nonlinear fourth-order parabolic system. SIAM J. Numer. Anal. 39 (2001) 385-406.
21. P.A. Markowich, Ch. A. Ringhofer and Ch. Schmeiser, Semiconductor Equations. First edition, Springer-Verlag, Wien (1990).
22. F. Pacard and A. Unterreiter, A variational analysis of the thermal equilibrium state of charged quantum fluids. Comm. Partial Differential Equations 20 (1995) 885-900.
23. P. Pietra and C. Pohl, Weak limits of the quantum hydrodynamic model. To appear in Proc. International Workshop on Quantum Kinetic Theory.
24. R. Pinnau, A note on boundary conditions for quantum hydrodynamic models. Appl. Math. Lett. 12 (1999) 77-82.
25. R. Pinnau, The linearized transient quantum drift diffusion model - stability of stationary states. ZAMM 80 (2000) 327-344.
26. R. Pinnau, Numerical study of the Quantum Euler-Poisson model. To appear in Appl. Math. Lett.
27. R. Pinnau and A. Unterreiter, The stationary current-voltage characteristics of the quantum drift diffusion model. SIAM J. Numer. Anal. 37 (1999) 211-245.
28. p(0, T; B). Ann. Mat. Pura Appl. 146 (1987) 65-96.
29. G.M. Troianiello, Elliptic Differential Equations and Obstacle Problems. First edition, Plenum Press, New York (1987).