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Building hierarchies of semiclassical Jacobi polynomials for spectral methods in annuli

Ioannis P. A. Papadopoulos, Timon S. Gutleb, Richard M. Slevinsky, Sheehan Olver

TL;DR

A sparse spectral element method by combining disk and annulus cells, which is highly effective for solving PDEs with radially discontinuous variable coefficients and data, and a particular application to constructing orthogonal polynomials in annuli is investigated.

Abstract

We discuss computing with hierarchies of families of (potentially weighted) semiclassical Jacobi polynomials which arise in the construction of multivariate orthogonal polynomials. In particular, we outline how to build connection and differentiation matrices with optimal complexity and compute analysis and synthesis operations in quasi-optimal complexity. We investigate a particular application of these results to constructing orthogonal polynomials in annuli, called the generalised Zernike annular polynomials, which lead to sparse discretisations of partial differential equations. We compare against a scaled-and-shifted Chebyshev--Fourier series showing that in general the annular polynomials converge faster when approximating smooth functions and have better conditioning. We also construct a sparse spectral element method by combining disk and annulus cells, which is highly effective for solving PDEs with radially discontinuous variable coefficients and data.

Building hierarchies of semiclassical Jacobi polynomials for spectral methods in annuli

TL;DR

A sparse spectral element method by combining disk and annulus cells, which is highly effective for solving PDEs with radially discontinuous variable coefficients and data, and a particular application to constructing orthogonal polynomials in annuli is investigated.

Abstract

We discuss computing with hierarchies of families of (potentially weighted) semiclassical Jacobi polynomials which arise in the construction of multivariate orthogonal polynomials. In particular, we outline how to build connection and differentiation matrices with optimal complexity and compute analysis and synthesis operations in quasi-optimal complexity. We investigate a particular application of these results to constructing orthogonal polynomials in annuli, called the generalised Zernike annular polynomials, which lead to sparse discretisations of partial differential equations. We compare against a scaled-and-shifted Chebyshev--Fourier series showing that in general the annular polynomials converge faster when approximating smooth functions and have better conditioning. We also construct a sparse spectral element method by combining disk and annulus cells, which is highly effective for solving PDEs with radially discontinuous variable coefficients and data.
Paper Structure (20 sections, 9 theorems, 78 equations, 11 figures, 3 algorithms)

This paper contains 20 sections, 9 theorems, 78 equations, 11 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Scaled-and-shifted Chebyshev--Fourier series
  3. Semiclassical Jacobi polynomials
  4. Connection matrices
  5. Jacobi matrices
  6. Analysis & synthesis
  7. Differentiation
  8. Computing the triple matrix products in $\mathcal{O}(N)$ complexity
  9. Orthogonal polynomials on the annulus
  10. Connection and differentiation matrices
  11. Analysis & synthesis
  12. Non-constant coefficients
  13. Sparse spectral method for PDEs
  14. Spectral element method
  15. Numerical examples
  16. ...and 5 more sections

Key Result

Theorem 3.2

\newlabelth:factorisations0 Let $w(x)$ denote a nonnegative and bounded weight on $\mathbb{R}$ and consider the measure $\mathrm{d}\mu$ such that both $\mathrm{d}\mu$ and $w(x) \mathrm{d}\mu$ are positive Borel measures on the real line whose support contains an infinite number of points and has f Here $R$ is upper-triangular, i.e. $R^\top R$ is a Cholesky factorisation. Moreover, if $w(x)$ is a

Figures (11)

  • Figure 1: Plots of the right-hand sides with discontinuities in the radial direction and the corresponding solution of \ref{['eq:spectral-poisson']}--\ref{['eq:spectral-helmholtz']}. The setup of the problem is given in \ref{['sec:disc-data']}. These functions are resolved to machine precision utilising the spectral element method described in \ref{['sec:spectral-element']} based on Zernike and Zernike annular polynomials.
  • Figure 1: (a) shows the $\mathcal{O}(N)$ complexity of the Cholesky as well as $Q$ and $R$ based QR methods for computing the $N \times N$ principal Jacobi matrix subblock of the semiclassical Jacobi polynomials $Q_n^{t,(1,1,22)}(x)$ from $Q_n^{t,(1,1,20)}(x)$. The Cholesky method requires twice the amount of work in this example as it proceeds in two steps. CPU timings were obtained on a Lenovo ThinkPad X1 Carbon laptop with a 12th Gen Intel i5-1250P CPU. (b) shows the relative 2-norm error for $N\times N$ principal subblocks of the Jacobi matrices comparing single to double precision computations.
  • Figure 1: The first three cells of a mesh for the spectral element method on a disk. The disk and annuli cells may vary in thickness.
  • Figure 1: Plots of the solutions of \ref{['eq:helmholtz']} with $\lambda(r) =80^2r^2$ and the right-hand side $f(x,y) = \sin(100x)$ on the annuli domains with inradii $\rho = 0.2$, $0.5$, and $0.8$, respectively (\ref{['sec:forced-helmholtz']}).
  • Figure 2: Spy plots of the matrices after discretising the Laplacian, $\Delta$, ((a) and (b)) and the Helmholtz operator $(\Delta + \mathcal{I})$ ((c) and (d)) on the 2D annulus $\Omega_{1/2}$ truncated at polynomial degree 20 with generalised Zernike annular polynomials. (a) and (c) show all the Fourier modes, whereas (b) and (d) show the matrix when one decouples the Fourier modes and focuses on the $m=0$ mode in isolation.
  • ...and 6 more figures

Theorems & Definitions (25)

  • Remark 1.1: Weak formulation
  • Remark 1.2
  • Remark 1.3: Generalisation to 3D
  • Remark 1.4: Generalisation to vector-valued PDEs
  • Remark 3.1
  • Theorem 3.2: Tab. 1 in Gutleb2023
  • Remark 3.3
  • Theorem 3.4: Theorem 2.20 in Gutleb2023
  • Remark 3.5
  • Definition 4.1: Generalised Zernike annular polynomials
  • ...and 15 more