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Inverse-average-type finite element discretizations of selfadjoint second-order elliptic problems

Authors: Peter A. Markowich and Miloš A. Zlámal
Journal: Math. Comp. 51 (1988), 431-449
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 second-order elliptic boundary value problems. The discretization differs from standard finite element methods by inverse-average-type approximations (along element sides) of the coefficient function $ a(x)$ in the operator $ - \operatorname{div}(a(x)\,{\operatorname{grad}}\,u)$. The derivation of the discretization is based on approximating the flux density $ J = a\,{\operatorname{grad}}{\mkern 1mu} u$ by constants on each element. In many cases the flux density is well behaved (moderately varying) even if $ a(x)$ and $ u(x)$ are fast varying.

Discretization methods of this type have been used successfully in semiconductor device simulation for many years; however, except in the one-dimensional case, the mathematical understanding of these methods was rather limited.

We analyze the stiffness matrix and prove that--under a rather mild restriction on the mesh--it 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 $ {H^1}$-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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  • [1] Robert A. Adams, Sobolev spaces, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Pure and Applied Mathematics, Vol. 65. MR 0450957
  • [2] I. Babuška and J. E. Osborn, Generalized finite element methods: their performance and their relation to mixed methods, SIAM J. Numer. Anal. 20 (1983), no. 3, 510–536. MR 701094,
  • [3] I. Babuška & J. E. Osborn, Finite Element Methods for the Solution of Problems with Rough Data, Lecture Notes in Math., Vol. 1121, Springer-Verlag, Berlin and New York, 1985, pp. 1-18.
  • [4] Philippe G. Ciarlet, The finite element method for elliptic problems, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. Studies in Mathematics and its Applications, Vol. 4. MR 0520174
  • [5] Pierre Grisvard, Behavior of the solutions of an elliptic boundary value problem in a polygonal or polyhedral domain, Numerical solution of partial differential equations, III (Proc. Third Sympos. (SYNSPADE), Univ. Maryland, College Park, Md., 1975) Academic Press, New York, 1976, pp. 207–274. MR 0466912
  • [6] B. Kawohl, Über nichtlineare gemischte Randwertprobleme für elliptische Differentialgleichungen zweiter Ordnung auf Gebieten mit Ecken, Dissertation, TH Darmstadt, BRD, 1978.
  • [7] Peter A. Markowich, The stationary semiconductor device equations, Computational Microelectronics, Springer-Verlag, Vienna, 1986. MR 821965
  • [8] M. S. Mock, Analysis of a discretization algorithm for stationary continuity equations in semiconductor device models. II, COMPEL 3 (1984), no. 3, 137–149. MR 782025,
  • [9] M. S. Mock, Analysis of a discretization algorithm for stationary continuity equations in semiconductor device models. II, COMPEL 3 (1984), no. 3, 137–149. MR 782025,
  • [10] Josef Nedoma, The finite element solution of parabolic equations, Apl. Mat. 23 (1978), no. 6, 408–438 (English, with Czech summary). With a loose Russian summary. MR 508545
  • [11] D. L. Scharfetter & H. K. Gummel, "Large signal analysis of a silicon read diode oscillator," IEEE Trans. Electron Devices, v. ED-16, 1969, pp. 64-77.
  • [12] S. Selberherr, Analysis and Simulation of Semiconductor Devices, Springer-Verlag, Wien and New York, 1984.
  • [13] Gilbert Strang and George J. Fix, An analysis of the finite element method, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1973. Prentice-Hall Series in Automatic Computation. MR 0443377
  • [14] W. V. Van Roosbroeck, "Theory of flow of electrons and holes in germanium and other semiconductors," Bell Syst. Techn. J., v. 29, 1950, pp. 560-607.
  • [15] Richard S. Varga, Matrix iterative analysis, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962. MR 0158502
  • [16] O. C. Zienkiewicz, The Finite Element Method, McGraw-Hill, London, 1977.
  • [17] M. A. Zlámal, "Finite element solution of the fundamental equations of semiconductor devices. II," submitted for publication, 1985.
  • [18] Miloš Zlámal, Finite element solution of the fundamental equations of semiconductor devices. I, Math. Comp. 46 (1986), no. 173, 27–43. MR 815829,

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