Error estimates of the quadrature finite element method with biquadratic finite elements for elliptic eigenvalue problems in the square domain

Abstract

A positive definite eigenvalue problem for second-order self-adjoint elliptic differential operator defined on a square domain in the plane with homogeneous Dirichlet boundary condition is considered. This problem has a nondecreasing sequence of positive eigenvalues of finite multiplicity with a limit point at infinity. To the sequence of eigenvalues, there corresponds an orthonormal system of eigenfunctions. The original differential eigenvalue problem is approximated by the finite element method with numerical integration and Lagrange biquadratic finite elements. Error estimates for approximate eigenvalues and eigenfunctions are established.