Preconditioning Legendre Spectral Collocation Methods for Elliptic Problems I: Finite Difference Operators

Abstract

In 1979 Orszag and Morchoisne independently proposed a finite-difference preconditioning of the Chebyshev collocation discretization of the Poisson equation. Over the years there has been intensive research, both experimental and theoretical, on the finite-difference and finite element preconditioning of both Chebyshev and Legendre spectral collocation methods for this problem. In this work we present the first mathematically rigorous results on finite-difference preconditioning of Legendre spectral collocation methods for the Helmholtz equation with Dirichlet boundary conditions. We show that there are two constants $0 < \Lambda_0 < \Lambda_1$ such that $$ {\rm Re } \{ (W_{n,m} A_{n,m} U,U)_{\ell_2} / (W_{n,m} L_{n,m} U,U)_{\ell_2} \} \geq \Lambda_0 $$ and $$ | ( W_{n,m} A_{n,m} U, U)_{\ell_2} / (W_{n,m} L_{n,m} U,U)_{\ell_2} ) | \leq \Lambda_1. $$ Here $W_{n,m} = {\rm diagonal} (w_k \tilde w_j)$, the diagonal matrix of Legendre--Gauss--Lobatto weights, $A_{n,m}$ is the collocation matrix, and $L_{n,m}$ is the finite difference operator. These results lead to the same estimates for the eigenvalues $\gamma_k$ of $L^{-1}_{n,m} A_{n,m}$. That is, $$ {\rm Re } \gamma_k \geq \Lambda_0 , $$ $$ | \gamma_k | \leq \Lambda_1 . $$