Preconditioning Chebyshev Spectral Collocation by Finite-Difference Operators

In 1979 Orszag proposed a finite-difference preconditioning of the Chebyshev collocation discretization of the Poisson equation. In 1984 Haldenwang, Labrosse, Abboudi, and DeVille gave analytic formulae for the eigenvalues of this preconditioned operator in the one-dimensional case. Experimental results over many years have shown the effectiveness of this procedure and appropriate bounds on the eigenvalues in two dimensions. However, there have been no mathematical proofs describing the behavior of the eigenvalues in two or more dimensions. In this work we consider the generalized field of values (U*ÂNU)/(U*LNU), where ÂNis the matrix of the Chebyshev collocation scheme and LNis the matrix of the finite-difference operator. For the case of the Chebyshev collocation of the Helmoltz operator, Au := -Δ u + au, a ≥ 0 preconditioned by the finite-difference operator associated with the Helmholtz operator Bu := -Δ u + bu, b ≥ 0, we prove that there are two constants$0 < \Lambda_0 < \Lambda_1$depending only on a(x, y) and b(x, y), but not on N, such that$\Re\{(U^\ast \hat{A}_N U)/(U^\ast L_N U)\} \geq \Lambda_0 > 0$and |(U*ÂNU)/(U*LNU)| ≤ Λ1. These results extend to higher dimensions and to bounds on |L-1 NÂN|1, wand |Â-1 NLN|1, win the general case where Au := -Δ u + a1ux+ a2uy+ au.