An asymptotic estimate of the error of eigenvalues in the method of finite elements for the Sturm-Liouville problem (Q1592089)

The Sturm-Liouville eigenvalue problem \[ (pu')'- qu+\lambda ru= 0,\quad 0< r< 1,\quad u(0)= u(1)= 0, \] is considered where by assumption \(p\) and \(r\) are positive functions, and \(q\) is a nonnegative function. Using a special basis consiting of solutions of related Cauchy problems, a second-order finite element scheme is constructed. Convergence of the method is proved, and an exact asymptotic estimate of the error in a computed eigenvalue is given.