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The ultraspherical rectangular collocation method and its convergence

Thomas Trogdon

TL;DR

The paper introduces the ultraspherical rectangular collocation (URC) method for linear two-point boundary-value problems on $[-1,1]$, and shows provable convergence when the leading coefficient equals one and collocation nodes are zeros of ultraspherical polynomials or Chebyshev variants. Building on the sparse ultraspherical method, URC combines a simple collocation framework with a rigorous preconditioning strategy to achieve practical $O(N^2)$ assembly and solution costs for GMRES, while outputting coefficient-space information that complements traditional value-space representations. The convergence theory blends operator preconditioning and polynomial-approximation arguments, yielding a main theorem that confirms spectral-rate convergence under full smoothness and near-spectral rates under finite smoothness; numerical experiments validate node-choice effects and preconditioning efficacy across a range of BVPs, including problems with boundary layers and non-smooth coefficients. The work advances a robust, efficient, and theoretically grounded tool for spectrally accurate collocation of high-order boundary-value problems, with implications for scalable solvers and flexible node selection in ultraspherical bases.

Abstract

We develop the ultraspherical rectangular collocation (URC) method, a collocation implementation of the sparse ultraspherical method of Olver \& Townsend for two-point boundary-value problems. The URC method is provably convergent, the implementation is simple and efficient, the convergence proof motivates a preconditioner for iterative methods, and the modification of collocation nodes is straightforward. The convergence theorem applies to all boundary-value problems when the coefficient functions are sufficiently smooth and when the roots of certain ultraspherical polynomials are used as collocation nodes. We also adapt a theorem of Krasnolsel'skii et al.~to our setting to prove convergence for the rectangular collocation method of Driscoll \& Hale for a restricted class of boundary conditions.

The ultraspherical rectangular collocation method and its convergence

TL;DR

The paper introduces the ultraspherical rectangular collocation (URC) method for linear two-point boundary-value problems on , and shows provable convergence when the leading coefficient equals one and collocation nodes are zeros of ultraspherical polynomials or Chebyshev variants. Building on the sparse ultraspherical method, URC combines a simple collocation framework with a rigorous preconditioning strategy to achieve practical assembly and solution costs for GMRES, while outputting coefficient-space information that complements traditional value-space representations. The convergence theory blends operator preconditioning and polynomial-approximation arguments, yielding a main theorem that confirms spectral-rate convergence under full smoothness and near-spectral rates under finite smoothness; numerical experiments validate node-choice effects and preconditioning efficacy across a range of BVPs, including problems with boundary layers and non-smooth coefficients. The work advances a robust, efficient, and theoretically grounded tool for spectrally accurate collocation of high-order boundary-value problems, with implications for scalable solvers and flexible node selection in ultraspherical bases.

Abstract

We develop the ultraspherical rectangular collocation (URC) method, a collocation implementation of the sparse ultraspherical method of Olver \& Townsend for two-point boundary-value problems. The URC method is provably convergent, the implementation is simple and efficient, the convergence proof motivates a preconditioner for iterative methods, and the modification of collocation nodes is straightforward. The convergence theorem applies to all boundary-value problems when the coefficient functions are sufficiently smooth and when the roots of certain ultraspherical polynomials are used as collocation nodes. We also adapt a theorem of Krasnolsel'skii et al.~to our setting to prove convergence for the rectangular collocation method of Driscoll \& Hale for a restricted class of boundary conditions.
Paper Structure (35 sections, 22 theorems, 183 equations, 6 figures)

This paper contains 35 sections, 22 theorems, 183 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Outline of paper, main results and relation to previous work
  3. Relation to previous results
  4. Notation
  5. Orthogonal polynomials
  6. Gaussian quadrature
  7. Interpolation
  8. Jacobi polynomials
  9. The Jacobi operator
  10. Ultraspherical (Gegenbauer) methods
  11. Differentiation
  12. Evaluation
  13. The URC method
  14. Connection coefficients (basis conversion)
  15. Function multiplication
  16. ...and 20 more sections

Key Result

Theorem 1.1

Let $t, \lambda > 0$, assume $a_k = 1$ and suppose that the roots of the $(k +\lambda)$th ultraspherical polynomials are used as collocation nodes. Then there exists $s,q> 0$ such that if $a_j \in C^q(\mathbb I)$, $j = 0,1,\ldots,k-1$, $f \in C^q(\mathbb I)$ and eq:bvp is uniquely solvable, then the

Figures (6)

  • Figure 1: The convergence of the URC method applied to \ref{['eq:second-order']}.
  • Figure 2: (A) The convergence of the URC method applied to \ref{['eq:third-order']}. (B) The solution with $N = 500$.
  • Figure 3: (A) The convergence of the URC method applied to \ref{['eq:boundary-layer']}. (B) The solution with $N = 1000$.
  • Figure 4: (A) The convergence of the URC method applied to \ref{['eq:boundary-layer']}. (B) The true solution. (C) The effect of varying $\lambda = 0,1/2,1$.
  • Figure 5: The convergence of the URC method applied to \ref{['eq:abs']} demonstrating that the convergence rate is determined by the smoothness of the true solution and not by the smoothness of the coefficient functions.
  • ...and 1 more figures

Theorems & Definitions (40)

  • Theorem 1.1: Informal
  • Theorem 1.2: Informal
  • Definition 1.3
  • Definition 2.1
  • Theorem 2.2
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • Lemma 3.3: Aliasing estimate
  • ...and 30 more