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Zbl 0552.65065
On the correction of finite difference eigenvalue approximations for Sturm-Liouville problems with general boundary conditions.
(English)
[J] BIT 24, 401-412 (1984). ISSN 0006-3835; ISSN 1572-9125/e

The error in the approximation to the kth eigenvalue of $-y''+qy=\lambda y,\alpha\sb 1y'(0)-\alpha\sb 2y(0)=\beta\sb 1y'(\pi)+\beta\sb 2y(\pi)=0,$ obtained by the standard centered difference method with step length h, is $O(k\sp 4h\sp 2)$. A major improvement was made by {\it J. W. Paine} and the authors [Computing 26, 123-139 (1981; Zbl 0436.65063)] who showed that, in the case $\alpha\sb 1=\beta\sb 1=0$, a simple correction reduced the error to $0(kh\sp 2)$. The present paper makes two further significant advances: the correction technique is extended to general $\alpha\sb 1$ and $\beta\sb 1$, and it is proved that the error in the corrected eigenvalues is $O(h\sp 2)$, i.e. it is independent of k. \par \{Reviewer's comments: 1. The asymptotic formulae, $\emptyset =O(k\sp{- 1}),$ ${\tilde \Phi}=O(k\sp{-1}),$ require the condition $\alpha\sb 1\ne 0$. 2. The role of the additional parameter $\alpha$ in the main theorem is clarified in a subsequent paper of the reviewer and {\it J. W. Paine} [Numer. Math. (to appear)] which examines a similar correction for Numerov's method.\}
[A.L.Andrew]
MSC 2000:
*65L15 Eigenvalue problems for ODE (numerical methods)
34L99 Ordinary differential operators

Keywords: finite element; finite difference eigenvalue; Sturm-Liouville problems; correction technique

Citations: Zbl 0436.65063

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