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Numerical solution of second-order elliptic equations on plane domains. (English) Zbl 0717.65082
The paper presents a general discretization method for linear convective diffusion equations. The schemes are based on an integral formula and have the following properties:
1. They are particularly suited to the case when convection is dominant;
2. Solutions obtained by them satisfy a discrete conservation law;
3. A discrete maximum principle is valid.
The author shows that the finite element solution converges to the exact one with the rate O(h) in \(H^ 1(D)\).

MSC:
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
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References:
[1] L. ANGERMANN, A mass-lumping semidiscretization of the semiconductor device equations. Part II : Error analysis. COMPEL 8 (1989), no. 2, 84-105. Zbl0687.65113 MR1020119 · Zbl 0687.65113
[2] [2] K. BABA, M. TABATA, On a conservative upwind finite element scheme for convective diffusion equations. R.A.I.R.O. Analyse numérique 15 (1981), no. 1, 3-25. Zbl0466.76090 MR610595 · Zbl 0466.76090
[3] R. E. BANK, D. J. ROSE, Some error estimates for the box method. SIAM J. Num. Anal. 24 (1987), no. 4, 777-787. Zbl0634.65105 MR899703 · Zbl 0634.65105
[4] P. CIARLET, The finite element method for elliptic problems. North-Holland Publishing Company, Amsterdam, New York - Oxford 1978. Zbl0383.65058 MR520174 · Zbl 0383.65058
[5] W. HACKBUSCH, On first and second order box schemes. Computing 41 (1989), 277-296. Zbl0649.65052 MR993825 · Zbl 0649.65052
[6] B. HEINRICH, Finite difference methods on irregular networks. Mathematical Reseach, vol. 33 Akademie-Verlag, Berlin 1987. Zbl0606.65065 MR1015930 · Zbl 0606.65065
[7] B. HEINRICH, Coercive and inverse-isotone discretization of diffusion-convection problems. AdW der DDR, Karl-Weierstrass-Institut fur Mathematik, PREPRINT P-MATH-19/88, Berlin 1988.
[8] T. IKEDA, Maximum pnnciple in finite element models for convection-diffusion phenomena. North-Holland Publishing Company, Amsterdam - New York -Oxford/Kmokumya Comp. Ltd., Tokyo 1983. Zbl0508.65049 MR683102 · Zbl 0508.65049
[9] B. J. MCCARTIN, J. R. CASPAR, R. E. LA BARRE, G. A. PETERSON, R. H. HOBBS, Steady state numerical analysis of single carrier two dimensional, semiconductor devices using the control area approximation. Proceedings of the NASECODE III conf, 185-190 Boole Press, Dublin 1983.
[10] J. J. H. MILLER, On the discretization of the semiconductor device equations in the two-dimensional case. Institute for Numerical and Computational Analysis, Preprint no. 1, Dublin 1986. Zbl0663.65128 · Zbl 0663.65128
[11] M. S. MOCK, On equations describing steady-state carrier distributions in a semiconductor device. Commun. Pure Appl. Math. 25 (1972), no. 6, 781-792. MR323233
[12] M. S. MOCK, Analysis of a discretization algorithm for stationary continuity equations in semiconductor device models, I-III. COMPEL 2 (1983), 117-139, 3 (1984), 137-149, 3 (1984), 187-199 Zbl0619.65117 MR782025 · Zbl 0619.65117
[13] U. RISCH, Die hybride upwind-FEM - ein einfaches Verfahren zur Behandlung konvektionsdominanter Randwertprobleme. Wiss Zeitschr. der TU Magdeburg 31 (1987), H 5, 88-94. Zbl0645.76095 MR951107 · Zbl 0645.76095
[14] H.-G. ROOS, Beziehungen zwischen Diskretisierungsverfahren fur Konvektions-Diffusions-Gleichungen und für die Grundgleichungen der inneren Elektronik. Informationen 07-10-86 der Technischen Universität, Dresden 1986.
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