Error investigation of finite element approximation for a nonlinear Sturm–Liouville problem

A positive definite differential eigenvalue problem with coefficients depending nonlinearly on the spectral parameter has been studied. The differential eigenvalue problem is formulated as a variational eigenvalue problem in a Hilbert space with bilinear forms nonlinearly depending on the spectral parameter. The variational problem has an increasing sequence of positive simple eigenvalues, which correspond to a normalized system of eigenfunctions. The variational problem has been approximated by a mesh scheme of the finite element method on the uniform grid with Lagrangian finite elements of arbitrary order. Error estimates for approximate eigenvalues and eigenfunctions in dependence on mesh size and eigenvalue size have been established. The obtained results are generalizations of the well-known results for differential eigenvalue problems with linear dependence on the spectral parameter.