Eigenvalue finite difference approximations for regular and singular Sturm-Liouville problems
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- by Nabil R. Nassif PDF
- Math. Comp. 49 (1987), 561-580 Request permission
Abstract:
This paper includes two parts. In the first part, general error estimates for "stable" eigenvalue approximations are obtained. These are practical in the sense that they are based on the discretization error of the difference formula over the eigenspace associated with the isolated eigenvalue under consideration. Verification of these general estimates are carried out on two difference schemes: that of Numerov to solve the Schrödinger singular equation and that of the central difference formula for regular Sturm-Liouville problems. In the second part, a sufficient condition for obtaining a "stable" difference scheme is derived. Such a condition (condition (N) of Theorem 2.1) leads to a simple "by hand" verification, when one selects a difference scheme to compute eigenvalues of a differential operator. This condition is checked for one- and two-dimensional problems.References
- J. H. Bramble and J. E. Osborn, Rate of convergence estimates for nonselfadjoint eigenvalue approximations, Math. Comp. 27 (1973), 525–549. MR 366029, DOI 10.1090/S0025-5718-1973-0366029-9 F. Chatelin, Théorie de l’Approximation des Opérateurs Linéaires, Application au Calcul des Valeurs Propres d’Opérateurs Différentiels et Intégraux, Lecture Notes, Grenoble University, 1977.
- Jean Descloux, Essential numerical range of an operator with respect to a coercive form and the approximation of its spectrum by the Galerkin method, SIAM J. Numer. Anal. 18 (1981), no. 6, 1128–1133. MR 639003, DOI 10.1137/0718077 J. Descloux, N. R. Nassif & J. Rappaz, "On spectral approximation, Part 1: The problem of convergence; Part 2: Error estimates for the Galerkin method," RAIRO Anal. Numér., v. 12, 1978, pp. 97-112; pp. 113-119.
- J. Descloux, N. Nassif, and J. Rappaz, On properties of spectral approximations, Equadiff IV (Proc. Czechoslovak Conf. Differential Equations and their Applications, Prague, 1977) Lecture Notes in Math., vol. 703, Springer, Berlin, 1979, pp. 81–85. MR 535326
- Rolf Dieter Grigorieff, Diskrete Approximation von Eigenwertproblemen. I. Qualitative Konvergenz, Numer. Math. 24 (1975), no. 4, 355–374 (German). MR 423099, DOI 10.1007/BF01397374
- Heinz-Otto Kreiss, Difference approximations for boundary and eigenvalue problems for ordinary differential equations, Math. Comp. 26 (1972), 605–624. MR 373296, DOI 10.1090/S0025-5718-1972-0373296-3 B. V. Numerov, Monthly Notices Roy. Astronom. Soc., v. 84, (180), 1924, p. 592.
- E. C. Titchmarsh, Eigenfunction expansions associated with second-order differential equations. Part I, 2nd ed., Clarendon Press, Oxford, 1962. MR 0176151
- John E. Osborn, Spectral approximation for compact operators, Math. Comput. 29 (1975), 712–725. MR 0383117, DOI 10.1090/S0025-5718-1975-0383117-3
- Gilbert Strang and George J. Fix, An analysis of the finite element method, Prentice-Hall Series in Automatic Computation, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1973. MR 0443377
- Friedrich Stummel, Elliptische Differenzenoperatoren unter Dirichletranbedingungen, Math. Z. 97 (1967), 169–211 (German). MR 224302, DOI 10.1007/BF01111697
- G. M. Vainikko, The difference method for ordinary differential equations, Ž. Vyčisl. Mat i Mat. Fiz. 9 (1969), 1057–1074 (Russian). MR 280027
- J. H. Wilkinson, The algebraic eigenvalue problem, Clarendon Press, Oxford, 1965. MR 0184422
Additional Information
- © Copyright 1987 American Mathematical Society
- Journal: Math. Comp. 49 (1987), 561-580
- MSC: Primary 65L10; Secondary 34B25
- DOI: https://doi.org/10.1090/S0025-5718-1987-0906189-1
- MathSciNet review: 906189