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

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 results on a frequently employed variant of finite element preconditioning of Legendre spectral collocation methods for the Helmholtz equation with Dirichlet boundary conditions. We show that there exist two positive constants $0 0 $$ and $$ |(W_{n,m} A_{n,m} U,U)_{\ell_2} / (W_{n,m} M^{-1}_{n,m} S_{n,m} U,U)_{\ell_2} | \leq \Lambda_1, $$ where $W_{n,m} = {\rm diagonal} (w_k \tilde w_j)$, the diagonal matrix of Legendre--Gauss--Lobatto (LGL) weights, and $A_{n,m}$ is the collocation matrix. $M_{n,m}$ is the mass matrix, and $S_{n,m}$ is the stiffness matrix of the finite element method. These estimates imply that the eigenvalues $\gamma_k$ of $S^{-1}_{n,m} M_{n,m} A_{n,m}$ satisfy $$ {\rm Re } \gamma_k \geq \Lambda_0 > 0, $$ $$ | \gamma_k | \leq \Lambda_1 . $$ These results depend on a thorough study of the ratio $2w_k / [ x_{k+1} - x_{k-1}]$, where $\{ x_k\}$ are LGL collocation points.