\input zb-basic
\input zb-ioport
\iteman{io-port 01608873}
\itemau{Parter, Seymour V.}
\itemti{Preconditioning Legendre spectral collocation methods for elliptic problems. I: Finite difference operators.}
\itemso{SIAM J. Numer. Anal. 39, No.1, 330-347 (2001).}
\itemab
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. By 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.
\itemrv{Gisbert Stoyan (Budapest)}
\itemcc{}
\itemut{Legendre spectral collocation; preconditioning; Poisson equation; Legendre-Gauss-Lobatto quadrature; Shortley-Weller difference operator; eigenvalues}
\itemli{doi:10.1137/S0036142999365060}
\end