Integration Preconditioning of Pseudospectral Operators. I. Basic Linear Operators

Abstract

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 √ 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.

Domain Decomposition Spectral Approximations for an Eigenvalue Problem with a Piecewise Constant Coefficient

“…The numerical computations are carried out for the case σ [12], which is chosen for the matrix K τ to be symmetric positive definite [9], [20]. The results, shown in Table 4, demonstrate the exponential rate of convergence.…”

Multidomain spectral method for the helically reduced wave equation

“…[40] considered general orthogonal polynomials, we work solely with Chebyshev polynomials, a classical system of orthogonal polynomials with many useful properties and applications [42,43]. Here we collect only those properties relevant for our discussion, mostly following [40,44,45]. The degree-n Chebyshev polynomial T n (ξ) is defined by T n (ξ) = cos(n arccos(ξ)) for −1 ≤ ξ ≤ 1, showing that we may consistently set T −n (ξ) = T n (ξ).…”

“…Moreover, most applications do not work directly with the analytic expansion coefficients (28). Rather, one typically approximates the integral in (28) via the quadrature rule stemming from Chebyshev-Gauss-Lobatto nodes (see, for example, [44]). This process introduces aliasing errors, and results in discrete expansion coefficients u discrete n .…”

Well-conditioned fractional collocation methods using fractional Birkhoff interpolation basis

“…However, the practicers are usually plagued with the dense, ill-conditioned linear systems, when compared with properly designed spectral-Galerkin approaches (see, e.g., [8,39]). The "local" finite-element preconditioners (see, e.g., [25]) and "global" integration preconditioners (see, e.g., [11,18,20,14,46,47]) were developed to overcome the ill-conditioning of the linear systems. When it comes to FDEs, it is advantageous to use collocation methods, as the Galerkin approaches usually lead to full dense matrices as well.…”