Correction of finite element eigenvalues for problems with natural or periodic boundary conditions

Abstract

Recent results of Andrew and Paine for a regular Sturm-Liouville problem with essential boundary conditions are extended to problems with natural or periodic boundary conditions. These results show that a simple asymptotic correction technique of Paine, de Hoog and Anderssen reduces the error in the estimate of thekth eigenvalue obtained by the finite element method, with linear hat functions and mesh lengthh, fromO(k4h2) toO(kh2). Numerical results show the correction to be useful even for low values ofk.