Convergent semidiscretization of a nonlinear fourth order parabolic system
ESAIM: Modélisation mathématique et analyse numérique, Tome 37 (2003) no. 2, pp. 277-289.

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.

DOI : 10.1051/m2an:2003026
Classification : 35K35, 65M12, 65M15, 65M20, 76Y05
Mots clés : higher order parabolic PDE, positivity, semidiscretization, stability, convergence, semiconductors
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     title = {Convergent semidiscretization of a nonlinear fourth order parabolic system},
     journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
     pages = {277--289},
     publisher = {EDP-Sciences},
     volume = {37},
     number = {2},
     year = {2003},
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     zbl = {1026.35045},
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Jüngel, Ansgar; Pinnau, René. Convergent semidiscretization of a nonlinear fourth order parabolic system. ESAIM: Modélisation mathématique et analyse numérique, Tome 37 (2003) no. 2, pp. 277-289. doi : 10.1051/m2an:2003026. http://www.numdam.org/articles/10.1051/m2an:2003026/

[1] R.A. Adams, Sobolev Spaces. First edition, Academic Press, New York (1975). | MR | Zbl

[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. | Zbl

[4] N. Ben Abdallah and A. Unterreiter, On the stationary quantum drift diffusion model. Z. Angew. Math. Phys. 49 (1998) 251-275. | Zbl

[5] F. Bernis and A. Friedman, Higher order nonlinear degenerate parabolic equations. J. Differential Equations 83 (1990) 179-206. | Zbl

[6] A.L. Bertozzi, The mathematics of moving contact lines in thin liquid films. Notices Amer. Math. Soc. 45 (1998) 689-697. | Zbl

[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. | Zbl

[8] A.L. Bertozzi and L. Zhornitskaya, Positivity preserving numerical schemes for lubriaction-typeequations. SIAM J. Numer. Anal. 37 (2000) 523-555. | Zbl

[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. | Zbl

[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. | Zbl

[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. | Zbl

[12] C.L. Gardner, The quantum hydrodynamic model for semiconductor devices. SIAM J. Appl. Math. 54 (1994) 409-427. | Zbl

[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. | Zbl

[14] I. Gasser and A. Jüngel, The quantum hydrodynamic model for semiconductors in thermal equilibrium. Z. Angew. Math. Phys. 48 (1997) 45-59. | Zbl

[15] I. Gasser and P.A. Markowich, Quantum hydrodynamics, Wigner transform and the classical limit. Asymptot. Anal. 14 (1997) 97-116. | Zbl

[16] G. Grün and M. Rumpf, Nonnegativity preserving convergent schemes for the thin film equation. Numer. Math. 87 (2000) 113-152. | Zbl

[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). | MR | Zbl

[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. | Zbl

[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. | Zbl

[21] P.A. Markowich, Ch. A. Ringhofer and Ch. Schmeiser, Semiconductor Equations. First edition, Springer-Verlag, Wien (1990). | Zbl

[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. | Zbl

[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. | Zbl

[25] R. Pinnau, The linearized transient quantum drift diffusion model - stability of stationary states. ZAMM 80 (2000) 327-344. | Zbl

[26] R. Pinnau, Numerical study of the Quantum Euler-Poisson model. To appear in Appl. Math. Lett. | Zbl

[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. | Zbl

[28] J. Simon, Compact sets in the space L p (0,T;B). Ann. Mat. Pura Appl. 146 (1987) 65-96. | Zbl

[29] G.M. Troianiello, Elliptic Differential Equations and Obstacle Problems. First edition, Plenum Press, New York (1987). | MR | Zbl

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