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A Numerical Approximation Structured by Mixed Finite Element and Upwind Fractional Step Difference for Semiconductor Device with Heat Conduction and Its Numerical Analysis

Part of: Partial differential equations, initial value and time-dependent initial-boundary value problems Applications to specific types of physical systems Partial differential equations, boundary value problems

Published online by Cambridge University Press:  20 June 2017

Yirang Yuan ,
Qing Yang ,
Changfeng Li  and
Tongjun Sun
Yirang Yuan*
Affiliation:
Institute of Mathematics, Shandong University, Jinan, 250100, China
Qing Yang
Affiliation:
School of Mathematical Sciences, Shandong Normal University, Jinan, 250014, China
Changfeng Li
Affiliation:
School of Economics, Shandong University, Jinan, 250100, China
Tongjun Sun
Affiliation:
Institute of Mathematics, Shandong University, Jinan, 250100, China
*
*Corresponding author. Email address:yryuan@sdu.edu.cn (Y. R. Yuan)
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Abstract

A coupled mathematical system of four quasi-linear partial differential equations and the initial-boundary value conditions is presented to interpret transient behavior of three dimensional semiconductor device with heat conduction. The electric potential is defined by an elliptic equation, the electron and hole concentrations are determined by convection-dominated diffusion equations and the temperature is interpreted by a heat conduction equation. A mixed finite element approximation is used to get the electric field potential and one order of computational accuracy is improved. Two concentration equations and the heat conduction equation are solved by a fractional step scheme modified by a second-order upwind difference method, which can overcome numerical oscillation, dispersion and computational complexity. This changes the computation of a three dimensional problem into three successive computations of one-dimensional problem where the method of speedup is used and the computational work is greatly shortened. An optimal second-order error estimate in L2 norm is derived by prior estimate theory and other special techniques of partial differential equations. This type of parallel method is important in numerical analysis and is most valuable in numerical application of semiconductor device and it can successfully solve this international famous problem.

Keywords

Three dimensional transient behavior of heat conduction problemnumerical simulation computationmixed finite elementmodified upwind fractional step differencesecond-order error estimates in L2 norm

MSC classification

Secondary: 65M60: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods 65N30: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods 82D37: Semiconductors
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 10 , Issue 3 , August 2017 , pp. 541 - 561
Copyright
Copyright © Global-Science Press 2017 

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