Finite element approximation to initialboundary value problems of the semiconductor device equations with magnetic influence
Author:
Jiang Zhu
Journal:
Math. Comp. 59 (1992), 3962
MSC:
Primary 65N30; Secondary 65N12
MathSciNet review:
1134742
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Abstract: We shall consider Zlámal's approach to the nonstationary equations of the semiconductor device theory under magnetic fields, with mixed boundary conditions. Owing to the reduced smoothness of the electric potential and carrier densities n and p caused by considering the mixed boundary conditions, we must use a nonstandard analysis for this procedure. Existence as well as uniqueness of the approximate solution is proved. The convergence rates obtained in this paper are slower than those previously obtained for pure Dirichlet or Neumann boundary conditions.
 [1]
W. Allegretto, Y. S. Mun, A. Nathan, and H. P. Baltes, Optimization of semiconductor magnetic field sensors using finite element analysis, Proc. NASECODE IV Conf., Boole Press, Dublin, 1985, pp. 129133.
 [2]
J.
Banasiak and G.
F. Roach, On mixed boundary value problems of Dirichlet
obliquederivative type in plane domains with piecewise differentiable
boundary, J. Differential Equations 79 (1989),
no. 1, 111–131. MR 997612
(90g:35044), http://dx.doi.org/10.1016/00220396(89)901162
 [3]
R. E. Bank, W. M. Fichtner, Jr., D. J. Rose, and R. K. Smith, Transient simulation of silicon devices and circuits, IEEE Trans. ComputerAided Design 4 (1985), 436451.
 [4]
Randolph
E. Bank, Joseph
W. Jerome, and Donald
J. Rose, Analytical and numerical aspects of semiconductor device
modeling, Computing methods in applied sciences and engineering, V
(Versailles, 1981), NorthHolland, Amsterdam, 1982,
pp. 593–597. MR 784655
(86h:78021)
 [5]
Philippe
G. Ciarlet, The finite element method for elliptic problems,
NorthHolland Publishing Co., AmsterdamNew YorkOxford, 1978. Studies in
Mathematics and its Applications, Vol. 4. MR 0520174
(58 #25001)
 [6]
Jim
Douglas Jr., Richard
E. Ewing, and Mary
Fanett Wheeler, The approximation of the pressure by a mixed method
in the simulation of miscible displacement, RAIRO Anal. Numér.
17 (1983), no. 1, 17–33 (English, with French
summary). MR
695450 (84f:76047)
 [7]
Jim
Douglas Jr., Irene
MartínezGamba, and M.
Cristina J. Squeff, Simulation of the transient behavior of a
onedimensional semiconductor device, Mat. Apl. Comput.
5 (1986), no. 2, 103–122 (English, with
Portuguese summary). MR 884996
(88d:81132)
 [8]
Jim
Douglas Jr. and Yuan
Yirang, Finite difference methods for the transient behavior of a
semiconductor device, Mat. Apl. Comput. 6 (1987),
no. 1, 25–37 (English, with Portuguese summary). MR 903000
(88h:65175)
 [9]
J. Douglas, Jr., Yirang Yuan, and Gang Li, A modified method of characteristic procedure for the transient behavior of a semiconductor device (preprint).
 [10]
, A mixed method for the transient behavior of a semiconductor device (preprint).
 [11]
Richard
E. Ewing, Timestepping Galerkin methods for nonlinear Sobolev
partial differential equations, SIAM J. Numer. Anal.
15 (1978), no. 6, 1125–1150. MR 512687
(80b:65136), http://dx.doi.org/10.1137/0715075
 [12]
R. E. Ewing and M. F. Wheeler, Galerkin methods for miscible displacement problems with point sources and sinksunit mobility ratio case, Lectures on the Numerical Solution of Partial Differential Equations, University of Maryland, 1981, pp. 151174.
 [13]
H.
Gajewski, On existence, uniqueness and asymptotic behavior of
solutions of the basic equations for carrier transport in
semiconductors, Z. Angew. Math. Mech. 65 (1985),
no. 2, 101–108 (English, with German and Russian summaries). MR 841263
(87k:35216), http://dx.doi.org/10.1002/zamm.19850650210
 [14]
H.
Gajewski and K.
Gröger, On the basic equations for carrier transport in
semiconductors, J. Math. Anal. Appl. 113 (1986),
no. 1, 12–35. MR 826656
(87i:82111), http://dx.doi.org/10.1016/0022247X(86)903306
 [15]
Irene
MartínezGamba and Maria
Cristina J. Squeff, Simulation of the transient behavior of a
onedimensional semiconductor device. II, SIAM J. Numer. Anal.
26 (1989), no. 3, 539–552. MR 997655
(90d:65219), http://dx.doi.org/10.1137/0726032
 [16]
Vivette
Girault and PierreArnaud
Raviart, Finite element methods for NavierStokes equations,
Springer Series in Computational Mathematics, vol. 5, SpringerVerlag,
Berlin, 1986. Theory and algorithms. MR 851383
(88b:65129)
 [17]
Joseph
W. Jerome, Consistency of semiconductor modeling: an
existence/stability analysis for the stationary Van Roosbroeck system,
SIAM J. Appl. Math. 45 (1985), no. 4, 565–590.
MR 796097
(87j:82014), http://dx.doi.org/10.1137/0145034
 [18]
, Evolution systems in semiconductor modeling: A cyclic uncoupled analysis for the Gummel map (to appear).
 [19]
Peter
A. Markowich, A singular perturbation analysis of the fundamental
semiconductor device equations, SIAM J. Appl. Math.
44 (1984), no. 5, 896–928. MR 759704
(86e:78024), http://dx.doi.org/10.1137/0144064
 [20]
, The stationary semiconductor device equations, SpringerVerlag, WienNew York, 1985.
 [21]
Peter
A. Markowich and C.
A. Ringhofer, A singularly perturbed boundary value problem
modelling a semiconductor device, SIAM J. Appl. Math.
44 (1984), no. 2, 231–256. MR 739302
(85k:34031), http://dx.doi.org/10.1137/0144018
 [22]
Peter
A. Markowich and Miloš
A. Zlámal, Inverseaveragetype finite element
discretizations of selfadjoint secondorder elliptic problems, Math. Comp. 51 (1988), no. 184, 431–449. MR 930223
(89a:65171), http://dx.doi.org/10.1090/S00255718198809302237
 [23]
M.
S. Mock, On equations describing steadystate carrier distributions
in a semiconductor device, Comm. Pure Appl. Math. 25
(1972), 781–792. MR 0323233
(48 #1591)
 [24]
M.
S. Mock, An initial value problem from semiconductor device
theory, SIAM J. Math. Anal. 5 (1974), 597–612.
MR
0417573 (54 #5623)
 [25]
Michael
S. Mock, Analysis of mathematical models of semiconductor
devices, Advances in Numerical Computation Series, vol. 3, Boole
Press, Dún Laoghaire, 1983. MR 697094
(84m:78002)
 [26]
C.
Ringhofer and C.
Schmeiser, An approximate Newton method for the solution of the
basic semiconductor device equations, SIAM J. Numer. Anal.
26 (1989), no. 3, 507–516. MR 997653
(90d:65220), http://dx.doi.org/10.1137/0726030
 [27]
S. Selberherr and C. Ringhofer, Discretization methods for the semiconductor equations, Proc. NASECODE III Conf., Boole Press, Dublin, 1983, pp. 3145.
 [28]
E.
Stephan and J.
R. Whiteman, Singularities of the Laplacian at corners and edges of
threedimensional domains and their treatment with finite element
methods, Math. Methods Appl. Sci. 10 (1988),
no. 3, 339–350. MR 949661
(89e:65124), http://dx.doi.org/10.1002/mma.1670100309
 [29]
Vidar
Thomée, Galerkin finite element methods for parabolic
problems, Lecture Notes in Mathematics, vol. 1054,
SpringerVerlag, Berlin, 1984. MR 744045
(86k:65006)
 [30]
Yuanming Wang, Mathematical model and its analysis for the carrier transport in semiconductor devices, Appl. Math. J. Chinese Univ. 2 (1987), 228240.
 [31]
Jiang
Zhu, Finite element methods for nonlinear Sobolev equations,
Northeast. Math. J. 5 (1989), no. 2, 179–196
(Chinese, with English summary). MR 1034130
(91c:65062)
 [32]
Jiang Zhu, Finite difference methods for the semiconductor device equations with magnetic influence (to appear).
 [33]
Miloš
Zlámal, Finite element solution of the
fundamental equations of semiconductor devices. I, Math. Comp. 46 (1986), no. 173, 27–43. MR 815829
(87d:65139), http://dx.doi.org/10.1090/S00255718198608158296
 [34]
, Finite element solution of the fundamental equations of semiconductor devices. II (to appear).
 [1]
 W. Allegretto, Y. S. Mun, A. Nathan, and H. P. Baltes, Optimization of semiconductor magnetic field sensors using finite element analysis, Proc. NASECODE IV Conf., Boole Press, Dublin, 1985, pp. 129133.
 [2]
 J. Banasiak and G. F. Roach, On mixed boundary value problems of Dirichlet obliquederivative type in plane domains with piecewise differentiable boundary, J. Differential Equations 79 (1989), 111131. MR 997612 (90g:35044)
 [3]
 R. E. Bank, W. M. Fichtner, Jr., D. J. Rose, and R. K. Smith, Transient simulation of silicon devices and circuits, IEEE Trans. ComputerAided Design 4 (1985), 436451.
 [4]
 R. E. Bank, J. W. Jerome, and D. J. Rose, Analytical and numerical aspects of semiconductor device modeling, Proc. Fifth Internat. Conf. on Computing Methods in Applied Science and Engineering (R. Glowinski and J. L. Lions, eds.), NorthHolland, Amsterdam, 1982, pp. 593597. MR 784655 (86h:78021)
 [5]
 P. G. Ciarlet, The finite element method for elliptic problems, NorthHolland, Amsterdam, New York, Oxford, 1978. MR 0520174 (58:25001)
 [6]
 J. Douglas, Jr., R. E. Ewing, and M. F. Wheeler, The approximation of the pressure by a mixed method in the simulation of miscible displacement, RAIRO Anal. Numer. 17 (1983), 1734. MR 695450 (84f:76047)
 [7]
 J. Douglas, Jr., I. M. Gamba, and M. C. J. Squeff, Simulation of the transient behavior of a onedimensional semiconductor device, Mat. Apl. Comput. 5 (1986), 103122. MR 884996 (88d:81132)
 [8]
 J. Douglas, Jr. and Yirang Yuan, Finite difference methods for the transient behavior of a semiconductor device, Mat. Apl. Comput. 6 (1987), 2538. MR 903000 (88h:65175)
 [9]
 J. Douglas, Jr., Yirang Yuan, and Gang Li, A modified method of characteristic procedure for the transient behavior of a semiconductor device (preprint).
 [10]
 , A mixed method for the transient behavior of a semiconductor device (preprint).
 [11]
 R. E. Ewing, Timestepping Galerkin methods for nonlinear Sobolev partial differential equations, SIAM J. Numer. Anal. 15 (1978), 11251150. MR 512687 (80b:65136)
 [12]
 R. E. Ewing and M. F. Wheeler, Galerkin methods for miscible displacement problems with point sources and sinksunit mobility ratio case, Lectures on the Numerical Solution of Partial Differential Equations, University of Maryland, 1981, pp. 151174.
 [13]
 H. Gajewski, On existence, uniqueness and asymptotic behavior of solutions of the basic equations for carrier transport in semiconductors, Z. Angew. Math. Mech. 65 (1985), 101108. MR 841263 (87k:35216)
 [14]
 H. Gajewski and K. Gröger, On the basic equations for carrier transport in semiconductors, J. Math. Anal. Appl. 113 (1986), 1235. MR 826656 (87i:82111)
 [15]
 I. M. Gamba and M. C. J. Squeff, Simulation of the transient behavior of a onedimensional semiconductor device. II, SIAM J. Numer. Anal. 26 (1989), 539552. MR 997655 (90d:65219)
 [16]
 V. Girault and P.A. Raviart, Finite element methods for NavierStokes equations: Theory and algorithms, Springer Ser. Comput. Math., vol. 5, SpringerVerlag, Berlin, Heidelberg, New York, Tokyo, 1986. MR 851383 (88b:65129)
 [17]
 J. W. Jerome, Consistency of semiconductor modeling: An existence/tability analysis for the stationary Van Roosbroeck system, SIAM J. Appl. Math. 45 (1985), 565590. MR 796097 (87j:82014)
 [18]
 , Evolution systems in semiconductor modeling: A cyclic uncoupled analysis for the Gummel map (to appear).
 [19]
 P. A. Markowich, A singular perturbation analysis of the fundamental semiconductor device equations, SIAM J. Appl. Math. 44 (1984), 896928. MR 759704 (86e:78024)
 [20]
 , The stationary semiconductor device equations, SpringerVerlag, WienNew York, 1985.
 [21]
 P. A. Markowich and C. Ringhofer, A singularly perturbed boundary value problem modelling a semiconductor device, SIAM J. Appl. Math. 44 (1984), 231256. MR 739302 (85k:34031)
 [22]
 P. A. Markowich and M. A. Zlámal, Inverseaveragetype finite element discretizations of selfadjoint secondorder elliptic problems, Math. Comp. 51 (1988), 431449. MR 930223 (89a:65171)
 [23]
 M. S. Mock, On equations describing steadystate carrier distributions in a semiconductor device, Comm. Pure Appl. Math. 25 (1972), 781792. MR 0323233 (48:1591)
 [24]
 , An initial value problem from semiconductor device theory, SIAM J. Math. Anal. 5 (1974), 597612. MR 0417573 (54:5623)
 [25]
 , Analysis of mathematical models of semiconductor devices, Boole Press, Dublin, 1983. MR 697094 (84m:78002)
 [26]
 C. Ringhofer and C. Schmeiser, An approximate Newton method for the solution of the basic semiconductor device equations, SIAM J. Numer. Anal. 26 (1989), 507516. MR 997653 (90d:65220)
 [27]
 S. Selberherr and C. Ringhofer, Discretization methods for the semiconductor equations, Proc. NASECODE III Conf., Boole Press, Dublin, 1983, pp. 3145.
 [28]
 E. Stephan and J. R. Whiteman, Singularities of the Laplacian at corners and edges of threedimensional domains and their treatment with finite element methods, Math. Methods Appl. Sci. 10 (1988), 339350. MR 949661 (89e:65124)
 [29]
 V. Thomée, Galerkin finite element methods for parabolic problems, Lecture Notes in Math., vol. 1054, SpringerVerlag, 1984. MR 744045 (86k:65006)
 [30]
 Yuanming Wang, Mathematical model and its analysis for the carrier transport in semiconductor devices, Appl. Math. J. Chinese Univ. 2 (1987), 228240.
 [31]
 Jiang Zhu, The finite element methods for nonlinear Sobolev equation, Northeast. Math. J. 5 (1989), 179196. MR 1034130 (91c:65062)
 [32]
 Jiang Zhu, Finite difference methods for the semiconductor device equations with magnetic influence (to appear).
 [33]
 M. A. Zlámal, Finite element solution of the fundamental equations of semiconductor devices. I, Math. Comp. 46 (1986), 2743. MR 815829 (87d:65139)
 [34]
 , Finite element solution of the fundamental equations of semiconductor devices. II (to appear).
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718199211347423
PII:
S 00255718(1992)11347423
Keywords:
Finite element approximation,
semiconductor device equation,
magnetic field
Article copyright:
© Copyright 1992
American Mathematical Society
