TY - JOUR

T1 - Convergence of a finite element method for the drift-diffusion semiconductor device equations

T2 - The zero diffusion case

AU - Cockburn, Bernardo

AU - Triandaf, Ioana

N1 - Copyright:
Copyright 2016 Elsevier B.V., All rights reserved.

PY - 1992/10

Y1 - 1992/10

N2 - In this paper a new explicit finite element method for numerically solving the drift-diffusion semiconductor device equations is introduced and analyzed. The method uses a mixed finite element method for the approximation of the electric field. A finite element method using discontinuous finite elements is used to approximate the concentrations, which may display strong gradients. The use of discontinuous finite elements renders the scheme for the concentrations trivially conservative and fully parallelizable. In this paper we carry out the analysis of the model method (which employs a continuous piecewise-linear approximation to the electric field and a piecewise-constant approximation to the electron concentration) in a model problem, namely, the so-called unipolar case with the diffusion terms neglected. The resulting system of equations is equivalent to a conservation law whose flux, the electric field, depends globally on the solution, the concentration of electrons. By exploiting the similarities of this system with classical scalar conservation laws, the techniques to analyze the monotone schemes for conservation laws are adapted to the analysis of the new scheme. The scheme, considered as a scheme for the electron concentration, is shown to satisfy a maximum principle and to be total variation bounded. Its convergence to the unique weak solution is proven. Numerical experiments displaying the performance of the scheme are shown.

AB - In this paper a new explicit finite element method for numerically solving the drift-diffusion semiconductor device equations is introduced and analyzed. The method uses a mixed finite element method for the approximation of the electric field. A finite element method using discontinuous finite elements is used to approximate the concentrations, which may display strong gradients. The use of discontinuous finite elements renders the scheme for the concentrations trivially conservative and fully parallelizable. In this paper we carry out the analysis of the model method (which employs a continuous piecewise-linear approximation to the electric field and a piecewise-constant approximation to the electron concentration) in a model problem, namely, the so-called unipolar case with the diffusion terms neglected. The resulting system of equations is equivalent to a conservation law whose flux, the electric field, depends globally on the solution, the concentration of electrons. By exploiting the similarities of this system with classical scalar conservation laws, the techniques to analyze the monotone schemes for conservation laws are adapted to the analysis of the new scheme. The scheme, considered as a scheme for the electron concentration, is shown to satisfy a maximum principle and to be total variation bounded. Its convergence to the unique weak solution is proven. Numerical experiments displaying the performance of the scheme are shown.

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U2 - 10.1090/S0025-5718-1992-1145661-0

DO - 10.1090/S0025-5718-1992-1145661-0

M3 - Article

AN - SCOPUS:84968494613

VL - 59

SP - 383

EP - 401

JO - Mathematics of Computation

JF - Mathematics of Computation

SN - 0025-5718

IS - 200

ER -