On the condition number of some spectral collocation operators and their finite element preconditioning

Spectral collocation approximations based on Legendre-Gauss-Lobatto nodes is considered. The collocation method is settled in a variational form, starting from the weak formulation of the differential problem. Numerical approximations to first and second order operators are introduced and the behavior of their discrete eigenvalues is studied. A finite element preconditioner for second order problems is proposed. Several numerical results concerning the condition number of the preconditioned spectral matrices and the application to conjugate gradient iterations are reported.