Error estimates of the finite difference method for eigenvalue problems with nonlinear entrance of the spectral parameter

Abstract

A positive definite second order ordinary differential eigenvalue problem with coefficients nonlinearly depending on the spectral parameter is studied. This differential nonlinear eigenvalue problem has an increasing sequence of positive simple eigenvalues, which correspond to a normalized system of eigenfunctions. The original differential nonlinear eigenvalue problem is approximated by a mesh scheme of the finite difference method on the uniform grid. New error estimates for approximate eigenvalues and approximate eigenfunctions in dependence on mesh size and eigenvalue size are established. Obtained theoretical results are generalizations of well-known results for differential eigenvalue problems with linear dependence on the spectral parameter.