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Corrected finite difference eigenvalues of periodic Sturm-Liouville problems. (English) Zbl 0930.65089
Finite element approximation to eigenvalues of regular Sturm-Liouville equations $-y''+qy=\lambda y$ can be improved applying a technique proposed by {\it J. W. Paine}, {\it F. R. de Hoog} and {\it R. S. Anderssen} [Computing 26, 123-139 (1981; Zbl 0445.65087)] in the case of boundary conditions $y(0)=y(\pi)=0$. In the present article periodic boundary conditions $y(0)=y(\pi)$, $y'(0)=y'(\pi)$ are considered. The author shows that a proof similar to that given by {\it A. L. Andrew} [J. Aust. Math. Soc., Ser. B 30, No. 4, 460-469 (1989; Zbl 0676.65089)] can be used to prove that the correction technique applied to a finite difference scheme given by {\it G. Vanden Berghe}, {\it M. Van Daele} and {\it H. De Meyer} [Appl. Numer. Math. 18, No. 1-3, 69-78 (1995; Zbl 0834.65075)] reduces the error in the $k$-th eigenvalue estimate from $O(k^4h^2)$ to $O(kh^2)$, where $h$ is the uniform mesh length.

65L15Eigenvalue problems for ODE (numerical methods)
34L15Eigenvalues, estimation of eigenvalues, upper and lower bounds for OD operators
65L12Finite difference methods for ODE (numerical methods)
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