On hp-convergence of prolate spheroidal wave functions and a new well-conditioned prolate-collocation scheme

Abstract

The first purpose of this paper is to provide a rigorous proof for the nonconvergence of h-refinement in hp-approximation by the PSWFs, a surprising convergence property that was first observed by Boyd et al [3, J. Sci. Comput., 2013]. The second purpose is to offer a new basis that leads to spectral-collocation systems with condition numbers independent of (c, N ), the intrinsic bandwidth parameter and the number of collocation points. In addition, this work gives insights into the development of effective spectral algorithms using this nonpolynomial basis. We in particular highlight that the collocation scheme together with a very practical rule for pairing up (c, N ) significantly outperforms the Legendre polynomialbased method (and likewise other Jacobi polynomial-based method) in approximating highly oscillatory bandlimited functions.1991 Mathematics Subject Classification. 65N35, 65E05, 65M70, 41A05, 41A10, 41A25.

Chebyshev operational matrix for solving fractional order delay-differential equations using spectral collocation method

“…The collocation method is the most common spectral methods used to solve differential equations, it is easy to implement once the differentiation matrices are computed. The coefficient matrix of the collocation system is always full with a condition number behaving like OðN 2m Þ (m is the order of the differential equation) see Du (2016), Shen, Tang, and Wang (2011), and Wang, Zhang, and Zhang (2014). In addition, if the exact solution exists then the error is estimated form:…”

A Review of Prolate Spheroidal Wave Functions from the Perspective of Spectral Methods

“…We then introduce the prolate points, quadrature rules, prolate interpolation and pseudospectral differentiation in a manner analogue to the polynomial-based spectral algorithms [77]. More importantly, we highlight the Kong-Rokhlin's rule in [47] for paring up (c,N), and present a stable way to compute the approximate "inverse" of the pseudospectral differentiation in [96], which can lead to a well-conditioned prolate-collocation method for BVPs. We reiterate that PSWFs are non-polynomials, lacking some important properties of orthogonal polynomials, so some care must be taken to build and assemble these pieces of the puzzle.…”

“…Wang et al [96] introduced a practical mean to implement this rule, which did not require computing the eigenvalues {λ N }. The essential idea was to replace λ N (c) by its tight explicit bound, which converted the problem of finding N * for given c and ε to locate the root of the algebraic equation:…”

“…Such a rule appears more reasonable as the PSWFs can best approximate the class of c-bandlimited functions. We remark that a more practice route to implement this rule can be found in [96].…”

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.…”