Convergent semidiscretization of a nonlinear fourth order parabolic system

Ansgar Jüngel; René Pinnau

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • 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 37.2 (2010): 277-289. <http://eudml.org/doc/194163>.

@article{Jüngel2010,
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},
keywords = {Higher order parabolic PDE; positivity; semidiscretization; stability; convergence; semiconductors.; semiconductors; implicit time discretization; backward Euler scheme},
language = {eng},
month = {3},
number = {2},
pages = {277-289},
publisher = {EDP Sciences},
title = {Convergent semidiscretization of a nonlinear fourth order parabolic system},
url = {http://eudml.org/doc/194163},
volume = {37},
year = {2010},
}

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
DA - 2010/3//
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.; semiconductors; implicit time discretization; backward Euler scheme
UR - http://eudml.org/doc/194163
ER -

References

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  1. R.A. Adams, Sobolev Spaces. First edition, Academic Press, New York (1975).  
  2. M.G. Ancona, Diffusion-drift modelling of strong inversion layers. COMPEL6 (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 Equations83 (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 Equations5 (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 Equations20 (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. ZAMM80 (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. J. Simon, Compact sets in the space Lp(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).  

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