Previous |  Up |  Next


semiconductor devices; finite element method; fully discrete approximate solution; convergence
In part I of the paper (see Zlámal [13]) finite element solutions of the nonstationary semiconductor equations were constructed. Two fully discrete schemes were proposed. One was nonlinear, the other partly linear. In this part of the paper we justify the nonlinear scheme. We consider the case of basic boundary conditions and of constant mobilities and prove that the scheme is unconditionally stable. Further, we show that the approximate solution, extended to the whole time interval as a piecewise linear function, converges in a strong norm to the weak solution of the semiconductor equations. These results represent an extended and corrected version of results announced without proof in Zlámal [14].
[1] J. Bergh, J. Löfström: Interpolation Spaces. Springer Verlag, Berlin-Heidelberg-New York, 1976. MR 0482275
[2] J. Céa: Optimization. Dunod, Paris, 1971. MR 0298892 | Zbl 0231.94026
[3] P. G. Ciarlet: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam-New York-Oxford, 1978. MR 0520174 | Zbl 0383.65058
[4] J. F. Ciavaldini: Analyse numérique d’un problème de Stefan à deux phases par une méthode d’eléments finis. SIAM J. Numer. Anal. 12 (1975), 464–487. DOI 10.1137/0712037 | MR 0391741 | Zbl 0272.65101
[5] D. Gilbarg, N. S. Trudinger: Elliptic Partial Differential Equations of Second Order. Springer-Verlag, Berlin-Heidelberg-New York, 1977. MR 0473443
[6] V. Giraud, P. A. Raviart: Finite Element Approximation of the Navier-Stokes Equations. Springer-Verlag, Berlin-Heidelberg-New York, 1979. MR 0540128
[7] G. Grisvard: Behaviour 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-San Francisco-London, 1978, pp. 207–274. MR 0466912
[8] A. Kufner, O. John and S. Fučík: Function Spaces. Academia, Prague, 1977. MR 0482102
[9] J. L. Lions: Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires. Dunod Gauthier-Villars, Paris, 1969. MR 0259693 | Zbl 0189.40603
[10] J. Nečas: Les Méthodes Directes en Théorie des Equations Elliptiques. Academia, Prague, 1967. MR 0227584
[11] R. Rannacher, R. Scott: Some optimal error estimates for piecewise linear finite element approximations. Math. Comp. 38 (1982), 437–445. DOI 10.1090/S0025-5718-1982-0645661-4 | MR 0645661
[12] P. A. Raviart: The use of numerical integration in finite element methods for solving parabolic equations. In: Topics in Numerical Analysis (J. J. H. Miller, ed.), Academic Press, London-New York, 1973, pp. 233–264. MR 0345428 | Zbl 0293.65086
[13] M. Zlámal: Finite element solution of the fundamental equations of semiconductor devices I. Math. Comp. 46 (1986), 27–43. DOI 10.1090/S0025-5718-1986-0815829-6 | MR 0815829
[14] M. Zlámal: Finite element solution of the fundamental equations of semiconductor devices. In: Numerical Approximation of Partial Differential Equations (E. L. Ortiz, ed.), North-Holland, Amsterdam-New York-Oxford-Tokyo, 1987, pp. 121–128. MR 0899784
[15] M. Zlámal: Curved elements in the finite element method I. SIAM J. Numer. Anal. 10 (1973), 229–240. DOI 10.1137/0710022 | MR 0395263
Partner of
EuDML logo