Correction of finite element estimates for Sturm-Liouville eigenvalues

Abstract

It is shown that a simple asymptotic correction technique of Paine, de Hoog and Anderssen reduces the error in the estimate of thekth eigenvalue of a regular Sturm-Liouville problem obtained by the finite element method, with linear hat functions and mesh lengthh, fromO(k 4 h 2) toO(k h 2). The result still holds when the matrix elements are evaluated by Simpson's rule, but if the trapezoidal rule is used the error isO(k 2 h 2). Numerical results demonstrate the usefulness of the correction even for low values ofk.