Error of the Finite Element Approximation for a Differential Eigenvalue Problem with Nonlinear Dependence on the Spectral Parameter

Abstract

The positive definite ordinary differential nonlinear eigenvalue problem of the second order with homogeneous Dirichlet boundary condition is considered. The problem is formulated as a symmetric variational eigenvalue problem with nonlinear dependence of the spectral parameter in a real infinite-dimensional Hilbert space. The variational eigenvalue problem consists in finding eigenvalues and corresponding eigenfunctions of the eigenvalue problem for a symmetric positive definite bounded bilinear form with respect to a symmetric positive definite completely continuous bilinear form in a real infinite-dimensional Hilbert space. The variational eigenvalue problem is approximated by the mesh scheme of the finite element method on the uniform grid. For constructing the mesh scheme, Lagrangian finite elements of arbitrary order are applied. Error estimates of approximate eigenvalues and error estimates of approximate eigenfunctions in the norm of initial real infinite-dimensional Hilbert space are established. These error estimates coincide in the order with error estimates of mesh scheme of the finite element method for linear eigenvalue problems. Moreover, superconvergence estimates for approximate eigenfunctions in the mesh norm with Gauss quadrature nodes are derived. Investigations of this paper generalize well known results for the eigenvalue problem with linear entrance on the spectral parameter.