Quadrature finite element method for the problem on eigenvibrations of a bar with elastic support

Abstract

The differential eigenvalue problem describing eigenvibrations of a bar with fixed ends and with elastic support at an interior point is investigated. This problem has an increasing sequence of positive simple eigenvalues with limit point at infinity. To the sequence of eigenvalues, there corresponds a complete orthonormal system of eigenfunctions. We formulate a limit differential eigenvalue problem and prove the convergence of the eigenvalues and eigenfunctions of the initial problem to the corresponding eigenvalues and eigenfunctions of the limit problem as stiffness coefficient tending to infinity. The original differential eigenvalue problem is approximated by the quadrature finite element method of arbitrary order on a uniform grid. Error estimates for approximate eigenvalues and eigenfunctions are established. Theoretical results are illustrated by numerical experiments for a model problem. Investigations of this paper can be generalized for the cases of more complicated and important problems on eigenvibrations of beams, plates and shells with elastic support.