Error estimates for a finite element method for the drift-diffusion semiconductor device equations: the zero diffusion case
Authors:
Bernardo Cockburn and Ioana Triandaf
Journal:
Math. Comp. 63 (1994), 51-76
MSC:
Primary 65M60; Secondary 35L60
DOI:
https://doi.org/10.1090/S0025-5718-1994-1226812-8
MathSciNet review:
1226812
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: In this paper new error estimates for an explicit finite element method for numerically solving the so-called zero-diffusion unipolar model (a one-dimensional simplified version of the drift-diffusion semiconductor device equations) are obtained. The method, studied in a previous paper, combines a mixed finite element method using a continuous piecewise-linear approximation of the electric field, with an explicit upwinding finite element method using a piecewise-constant approximation of the electron concentration. By using a suitable extension of Kuznetsov approximation theory for scalar conservation laws, it is proved that, under proper hypotheses on the data, the ${L^\infty }({L^1})$-error between the approximate and exact electron concentrations of the zero-diffusion unipolar model is of order $\Delta {x^{1/2}}$. These estimates are sharp.
- Bernardo Cockburn and Ioana Triandaf, Convergence of a finite element method for the drift-diffusion semiconductor device equations: the zero diffusion case, Math. Comp. 59 (1992), no. 200, 383–401, S29–S46. MR 1145661, DOI https://doi.org/10.1090/S0025-5718-1992-1145661-0
- Bernardo Cockburn, Quasimonotone schemes for scalar conservation laws. I, SIAM J. Numer. Anal. 26 (1989), no. 6, 1325–1341. MR 1025091, DOI https://doi.org/10.1137/0726077
- Bernardo Cockburn, Frédéric Coquel, and Philippe LeFloch, An error estimate for finite volume methods for multidimensional conservation laws, Math. Comp. 63 (1994), no. 207, 77–103. MR 1240657, DOI https://doi.org/10.1090/S0025-5718-1994-1240657-4
- Jim Douglas Jr., Irene Martínez-Gamba, and M. Cristina J. Squeff, Simulation of the transient behavior of a one-dimensional semiconductor device, Mat. Apl. Comput. 5 (1986), no. 2, 103–122 (English, with Portuguese summary). MR 884996 S. N. Kružkov, First order quasilinear equations in several independent variables, Math. USSR-Sb. 10 (1970), 217-243. N. N. Kuznetsov, Accuracy of some approximate methods for computing the weak solutions of a first-order quasi-linear equation, U.S.S.R. Comput. Math. and Math. Phys. 16 (1976), 105-119. A. Y. LeRoux, Etude du problème mixte pour une équation quasilinéaire du premier ordre, C. R. Acad. Sci. Paris Sér. A 285 (1977), 351-354.
- Bradley J. Lucier, Error bounds for the methods of Glimm, Godunov and LeVeque, SIAM J. Numer. Anal. 22 (1985), no. 6, 1074–1081. MR 811184, DOI https://doi.org/10.1137/0722064
- Bradley J. Lucier, A moving mesh numerical method for hyperbolic conservation laws, Math. Comp. 46 (1986), no. 173, 59–69. MR 815831, DOI https://doi.org/10.1090/S0025-5718-1986-0815831-4
- Bradley J. Lucier, On nonlocal monotone difference schemes for scalar conservation laws, Math. Comp. 47 (1986), no. 175, 19–36. MR 842121, DOI https://doi.org/10.1090/S0025-5718-1986-0842121-6
- Irene Martínez-Gamba and Maria Cristina J. Squeff, Simulation of the transient behavior of a one-dimensional semiconductor device. II, SIAM J. Numer. Anal. 26 (1989), no. 3, 539–552. MR 997655, DOI https://doi.org/10.1137/0726032
- Richard Sanders, On convergence of monotone finite difference schemes with variable spatial differencing, Math. Comp. 40 (1983), no. 161, 91–106. MR 679435, DOI https://doi.org/10.1090/S0025-5718-1983-0679435-6
- Michael Sever, Analysis of a discretization algorithm for time-dependent semiconductor models, COMPEL 6 (1987), no. 3, 171–189. MR 978490, DOI https://doi.org/10.1108/eb010033 V. W. Van Roosbroeck, Theory of flow of electrons and holes in germanium and other semiconductors, Bell Syst. Tech. J. 29 (1950), 560-607.
Retrieve articles in Mathematics of Computation with MSC: 65M60, 35L60
Retrieve articles in all journals with MSC: 65M60, 35L60
Additional Information
Keywords:
Semiconductor devices,
conservation laws,
finite elements,
convergence
Article copyright:
© Copyright 1994
American Mathematical Society