Preconditioning Legendre spectral collocation methods for elliptic problems. I: Finite difference operators (Q2719210)

The author investigates the preconditioning by the Shortley-Weller difference operator of the Legendre-collocation matrix arising from the representation of the approximate solution (of a first-kind boundary value problem for the 2D Poisson equation posed in the unit square) by a tensor-product Lagrange basis and collocating it at Gauss-Lobatto points. NEWLINENEWLINENEWLINEBy a careful analysis of the properties of the Gauss-Lobatto points, of the corresponding weights and of the 1D case, the author obtains estimates of the eigenvalues of the preconditioned matrix which show the eigenvalues to lie in a circle (of a radius which is bounded independently of the discretization parameters) around the origin, and to be in the right half plane, bounded away from the imaginary axis.