Inverseaveragetype finite element discretizations of selfadjoint secondorder elliptic problems
Authors:
Peter A. Markowich and Miloš A. Zlámal
Journal:
Math. Comp. 51 (1988), 431449
MSC:
Primary 65N30
MathSciNet review:
930223
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Abstract: This paper is concerned with the analysis of a class of "special purpose" piecewise linear finite element discretizations of selfadjoint secondorder elliptic boundary value problems. The discretization differs from standard finite element methods by inverseaveragetype approximations (along element sides) of the coefficient function in the operator . The derivation of the discretization is based on approximating the flux density by constants on each element. In many cases the flux density is well behaved (moderately varying) even if and are fast varying. Discretization methods of this type have been used successfully in semiconductor device simulation for many years; however, except in the onedimensional case, the mathematical understanding of these methods was rather limited. We analyze the stiffness matrix and prove thatunder a rather mild restriction on the meshit is a diagonally dominant Stieltjes matrix. Most importantly, we derive an estimate which asserts that the piecewise linear interpolant of the solution u is approximated to order 1 by the finite element solution in the norm. The estimate depends only on the mesh width and on derivatives of the flux density and of a possibly occurring inhomogeneity.
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 R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975. MR 0450957 (56:9247)
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 P. G. Ciarlet, The Finite Element Method for Elliptic Problems. NorthHolland, Amsterdam, 1978. MR 0520174 (58:25001)
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 P. Grisvard, "Behavior of the solutions of an elliptic boundary value problem in a polygonal or polyhedral domain," in Numerical Solution of Partial Differential Equations III (B. Hubbard, ed.), Academic Press, New York, 1976, pp. 207274. MR 0466912 (57:6786)
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 P. A. Markowich, The Stationary Semiconductor Device Equations, SpringerVerlag, Wien and New York, 1986. MR 821965 (87b:78042)
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 M. S. Mock, "Analysis of a discretization algorithm for stationary continuity equations in semiconductor device models. I," Compel, v. 2, 1983, pp. 117139. MR 782025 (86i:78013a)
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 D. L. Scharfetter & H. K. Gummel, "Large signal analysis of a silicon read diode oscillator," IEEE Trans. Electron Devices, v. ED16, 1969, pp. 6477.
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 S. Selberherr, Analysis and Simulation of Semiconductor Devices, SpringerVerlag, Wien and New York, 1984.
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 O. C. Zienkiewicz, The Finite Element Method, McGrawHill, London, 1977.
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 M. A. Zlámal, "Finite element solution of the fundamental equations of semiconductor devices. II," submitted for publication, 1985.
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 M. A. Zlámal, "Finite element solution of the fundamental equations of semiconductor devices. I," Math. Comp., v. 46, 1986, pp. 2743. MR 815829 (87d:65139)
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DOI:
http://dx.doi.org/10.1090/S00255718198809302237
PII:
S 00255718(1988)09302237
Article copyright:
© Copyright 1988
American Mathematical Society
