Correction of finite element estimates for Sturm-Liouville eigenvalues

Abstract

Numerical results demonstrate the usefulness of the correction even for low values of k.

Small potential corrections for the discrete eigenvalues of the Sturm-Liouville problem

“…for l<k<(n+l)/2) for the solution of the inverse eigenvalue problem (2), which gives the desired approximation for q. In this paper, the known result of [2] that Ak--Ak= O(kZh3/sin kh)= O(kh2), for ke [1, an] and some fixed ae(0, 1), is confirmed and modified. The result is given in the following main theorem, where llq]l,,(m~N) is the norm of q as an element of the Sobolev space Hr"(0, n).…”

“…The results are also based on new upper and lower bounds for 2k given in [5]. The theorem suggests that instead of 2k the corrected eigenvalues Ak(1 <k<=(n+ 1)/2) given by (3) can be used as a substitute for Ak for the solution of the inverse Sturm-Liouville Problem (1) via the Galerkin equations (2). The proof of Theorem 1 is presented in Sect.…”

“…Generally, the higher discrete eigenvalues of (2) in {A1, ..., A,} strongly deviate from the corresponding eigenvalues of (1) in {21 ..... 2,}. According to an idea of Paine, de Hoog and Anderssen [6] one introduces the differences/z k-k 2 of the corresponding eigenvalues of (2) and (1) for the case q=0 as asymptotic eigenvalue corrections for the 2k. More precisely, one takes (3) Ak "=2k + Pk --k z and uses A=Ak instead of A=2k (e.g.…”

“…More precisely, one takes (3) Ak "=2k + Pk --k z and uses A=Ak instead of A=2k (e.g. for l<k<(n+l)/2) for the solution of the inverse eigenvalue problem (2), which gives the desired approximation for q. In this paper, the known result of [2] that Ak--Ak= O(kZh3/sin kh)= O(kh2), for ke [1, an] and some fixed ae(0, 1), is confirmed and modified.…”

Matrix methods for computing eigenvalues of Sturm–Liouville problems of order four

“…However, the two most common matrix methods are finite different method (FDM) and Numerov's method [12]. Correction terms of the methods for second order SLP have been described in [12,13,14,15,16,17,18,19,20] to obtain better numerical eigenvalues of the problem.…”