Integration preconditioning of pseudospectral operators. I. Basic linear operators.

*(English)*Zbl 0912.65067Author’s abstract: This paper develops a family of preconditioners for pseudospectral approximations of \(p\)th-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 \(\sqrt N\), 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.

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 \(\sqrt N\), 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.

Reviewer: E.L.Allgower (Fort Collins)

##### MSC:

65L10 | Numerical solution of boundary value problems involving ordinary differential equations |

34L05 | General spectral theory of ordinary differential operators |

65M70 | Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs |

34B05 | Linear boundary value problems for ordinary differential equations |

65F35 | Numerical computation of matrix norms, conditioning, scaling |

65N35 | Spectral, collocation and related methods for boundary value problems involving PDEs |

65F10 | Iterative numerical methods for linear systems |