Integration preconditioning of pseudospectral operators. I. Basic linear operators

This paper develops a family of preconditioners for pseudospectral approximations of pth-order linear differential operators subject to various types of boundary conditions. The approximations are based on ultraspherical polynomials with special attention being paid to Legendre and Chebyshev polynomial methods based on Gauss-Lobatto quadrature points. The eigenvalue spectrum of the preconditioned operators are obtained in closed analytic form and the weakly enforced boundary conditions are shown to result in a rank 2p perturbation of the identity operator, i.e., the majority of the preconditioned eigenvalues are unity. The spectrum of the preconditioned advective operator is shown to be bounded independent of the order of the approximation, N. However, the preconditioned diffusive operator is, in general, indefinite with four real eigenvalues. For Dirichlet boundary conditions the spectral radius grows as root N, while it scales as N for the case of Neumann boundary conditions. These results are shown to be asymptotically optimal within the present framework. Generalizations to higher-order differential operators, general boundary conditions, and arbitrary polynomial basis and quadrature nodes are discussed.