Convergent semidiscretization of a nonlinear fourth order parabolic system

Ansgar Jüngel; René Pinnau

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2003)

  • Volume: 37, Issue: 2, page 277-289
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Jüngel, Ansgar, and Pinnau, René. "Convergent semidiscretization of a nonlinear fourth order parabolic system." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 37.2 (2003): 277-289. <http://eudml.org/doc/245295>.

@article{Jüngel2003,
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.},
author = {Jüngel, Ansgar, Pinnau, René},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {higher order parabolic PDE; positivity; semidiscretization; stability; convergence; semiconductors; implicit time discretization; backward Euler scheme},
language = {eng},
number = {2},
pages = {277-289},
publisher = {EDP-Sciences},
title = {Convergent semidiscretization of a nonlinear fourth order parabolic system},
url = {http://eudml.org/doc/245295},
volume = {37},
year = {2003},
}

TY - JOUR
AU - Jüngel, Ansgar
AU - Pinnau, René
TI - Convergent semidiscretization of a nonlinear fourth order parabolic system
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2003
PB - EDP-Sciences
VL - 37
IS - 2
SP - 277
EP - 289
AB - 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.
LA - eng
KW - higher order parabolic PDE; positivity; semidiscretization; stability; convergence; semiconductors; implicit time discretization; backward Euler scheme
UR - http://eudml.org/doc/245295
ER -

References

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  1. [1] R.A. Adams, Sobolev Spaces. First edition, Academic Press, New York (1975). Zbl0314.46030MR450957
  2. [2] M.G. Ancona, Diffusion–drift modelling of strong inversion layers. COMPEL 6 (1987) 11–18. 
  3. [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. Zbl0913.65084
  4. [4] N. Ben Abdallah and A. Unterreiter, On the stationary quantum drift diffusion model. Z. Angew. Math. Phys. 49 (1998) 251–275. Zbl0936.35057
  5. [5] F. Bernis and A. Friedman, Higher order nonlinear degenerate parabolic equations. J. Differential Equations 83 (1990) 179–206. Zbl0702.35143
  6. [6] A.L. Bertozzi, The mathematics of moving contact lines in thin liquid films. Notices Amer. Math. Soc. 45 (1998) 689–697. Zbl0917.35100
  7. [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. Zbl0916.35008
  8. [8] A.L. Bertozzi and L. Zhornitskaya, Positivity preserving numerical schemes for lubriaction–typeequations. SIAM J. Numer. Anal. 37 (2000) 523–555. Zbl0961.76060
  9. [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. Zbl0806.35059
  10. [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. Zbl0692.65067
  11. [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. Zbl0929.35061
  12. [12] C.L. Gardner, The quantum hydrodynamic model for semiconductor devices. SIAM J. Appl. Math. 54 (1994) 409–427. Zbl0815.35111
  13. [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. Zbl0957.76099
  14. [14] I. Gasser and A. Jüngel, The quantum hydrodynamic model for semiconductors in thermal equilibrium. Z. Angew. Math. Phys. 48 (1997) 45–59. Zbl0882.76108
  15. [15] I. Gasser and P.A. Markowich, Quantum hydrodynamics, Wigner transform and the classical limit. Asymptot. Anal. 14 (1997) 97–116. Zbl0877.76087
  16. [16] G. Grün and M. Rumpf, Nonnegativity preserving convergent schemes for the thin film equation. Numer. Math. 87 (2000) 113–152. Zbl0988.76056
  17. [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. Zbl1174.82348
  18. [18] A. Jüngel, Quasi-hydrodynamic Semiconductor Equations. Birkhäuser, PNLDE 41 (2001). Zbl0969.35001MR1818867
  19. [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. Zbl0979.35061
  20. [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. Zbl0994.35047
  21. [21] P.A. Markowich, Ch. A. Ringhofer and Ch. Schmeiser, Semiconductor Equations. First edition, Springer–Verlag, Wien (1990). Zbl0765.35001
  22. [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. Zbl0820.35112
  23. [23] P. Pietra and C. Pohl, Weak limits of the quantum hydrodynamic model. To appear in Proc. International Workshop on Quantum Kinetic Theory. 
  24. [24] R. Pinnau, A note on boundary conditions for quantum hydrodynamic models. Appl. Math. Lett. 12 (1999) 77–82. Zbl0952.76100
  25. [25] R. Pinnau, The linearized transient quantum drift diffusion model – stability of stationary states. ZAMM 80 (2000) 327–344. Zbl0947.35166
  26. [26] R. Pinnau, Numerical study of the Quantum Euler–Poisson model. To appear in Appl. Math. Lett. Zbl1077.82022
  27. [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. Zbl0981.65076
  28. [28] J. Simon, Compact sets in the space L p ( 0 , T ; B ) . Ann. Mat. Pura Appl. 146 (1987) 65–96. Zbl0629.46031
  29. [29] G.M. Troianiello, Elliptic Differential Equations and Obstacle Problems. First edition, Plenum Press, New York (1987). Zbl0655.35002MR1094820

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