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Inverse-average-type finite element discretizations of selfadjoint second-order elliptic problems. (English) Zbl 0699.65074
Summary: 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 $$-div(a(x)\text{grad} u).$$ The derivation of the discretization is based on approximating the flux density $$J=a \text{grad} u$$ 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.

##### MSC:
 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 35J25 Boundary value problems for second-order elliptic equations 78A55 Technical applications of optics and electromagnetic theory
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##### References:
 [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. · Zbl 0314.46030 [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. · Zbl 0528.65046 [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. · Zbl 0383.65058 [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. [6] B. Kawohl, Über nichtlineare gemischte Randwertprobleme für elliptische Differentialgleichungen zweiter Ordnung auf Gebieten mit Ecken, Dissertation, TH Darmstadt, BRD, 1978. · Zbl 0415.35030 [7] Peter A. Markowich, The stationary semiconductor device equations, Computational Microelectronics, Springer-Verlag, Vienna, 1986. · Zbl 0614.34013 [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. , https://doi.org/10.1108/eb009992 M. S. Mock, Analysis of a discretization algorithm for stationary equations in semiconductor device models. III, COMPEL 3 (1984), no. 4, 187 – 199. · Zbl 0619.65118 [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. , https://doi.org/10.1108/eb009992 M. S. Mock, Analysis of a discretization algorithm for stationary equations in semiconductor device models. III, COMPEL 3 (1984), no. 4, 187 – 199. · Zbl 0619.65118 [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. · Zbl 0427.65075 [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. · Zbl 0356.65096 [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. · Zbl 1372.35295 [15] Richard S. Varga, Matrix iterative analysis, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962. · Zbl 0133.08602 [16] O. C. Zienkiewicz, The Finite Element Method, McGraw-Hill, London, 1977. · Zbl 0435.73072 [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. · Zbl 0609.65089
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