Finite element and NURBS approximations of eigenvalue, boundary-value, and initial-value problems

We study the spectral approximation properties of finite element and NURBS spaces from a global perspective. We focus on eigenfunction approximations and discover that the L2- norm errors for finite element eigenfunctions exhibit pronounced ''spikes'' about the transition points between branches of the eigenvalue spectrum. This pathology is absent in NURBS approximations. By way of the Pythagorean eigenvalue error theorem, we determine that the squares of the energy-norm errors of the eigenfunctions are the sums of the eigenvalue errors and the squares of the L2-norm eigenfunction errors. The spurious behavior of the higher eigenvalues for standard finite elements is well-known and therefore inherited by the energy-norm errors along with the spikes in the L2-norm of the eigenfunction errors. The eigenvalue pathology is absent for NURBS. The implications of these results to the corresponding elliptic boundary-value problem and parabolic and hyperbolic initial-value problems are discussed.