Preconditioning Chebyshev Spectral Collocation Method for Elliptic Partial Differential Equations

Abstract

In this paper we analyze a preconditioning technique for the solution of Chebyshev spectral collocation equations with Dirichlet boundary conditions. We obtain bounds on the eigenvalues for the Helmholtz equation. These eigenvalue bounds are obtained as a consequence of estimates on the field of values $(\tilde A_{N^2}U,U)_{l_2}/(\tilde Q_{N^2}U,U)_{l_2}$, where $\tilde A_{N^2}$ is the weighted collocation matrix and $\tilde Q_{N^2}$ is the preconditioner. The preconditioner $\tilde Q_{N^2}$ is robust in the sense that it provides bounds on the $H^1_{0,w}$ condition number of $\tilde Q_{N^2}^{-1}\tilde L_{N^2}$ when $\tilde L_{N^2}$ is the weighted collocation matrix associated with the general elliptic operator $Lu:= -\Delta u + a_1u_x + a_2u_y + a_0u$.