Corrected finite difference eigenvalues of periodic Sturm-Liouville problems (Q1294609)

Finite element approximation to eigenvalues of regular Sturm-Liouville equations \(-y''+qy=\lambda y\) can be improved applying a technique proposed by \textit{J. W. Paine}, \textit{F. R. de Hoog} and \textit{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 \textit{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 \textit{G. Vanden Berghe}, \textit{M. Van Daele} and \textit{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.