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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 $-{y}^{\text{'}\text{'}}+qy=\lambda y$ can be improved applying a technique proposed by J. W. Paine, F. R. de Hoog and R. S. Anderssen [Computing 26, 123-139 (1981; Zbl 0445.65087)] in the case of boundary conditions $y\left(0\right)=y\left(\pi \right)=0$. In the present article periodic boundary conditions $y\left(0\right)=y\left(\pi \right)$, ${y}^{\text{'}}\left(0\right)={y}^{\text{'}}\left(\pi \right)$ are considered. The author shows that a proof similar to that given by 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 G. Vanden Berghe, M. Van Daele and 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 $O\left({k}^{4}{h}^{2}\right)$ to $O\left(k{h}^{2}\right)$, where $h$ is the uniform mesh length.
##### MSC:
 65L15 Eigenvalue problems for ODE (numerical methods) 34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds for OD operators 65L12 Finite difference methods for ODE (numerical methods)