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A fully diagonalized spectral method on the unit ball

Miguel A. Piñar

TL;DR

This work develops a fully diagonalized spectral method for Dirichlet boundary-value problems on the unit ball by leveraging Sobolev orthogonal polynomials. It builds a basis from spherical harmonics and univariate radial Sobolev polynomials, enabling direct computation of Sobolev-Fourier coefficients via a connection to classical ball polynomials and avoiding large linear systems. The approach introduces the univariate Sobolev polynomials $q_j^{(\beta)}$ and their connection to Jacobi polynomials, provides recursive schemes to obtain the needed coefficients, and demonstrates exponential convergence in a 2D numerical experiment. The method offers a scalable, high-accuracy alternative for spectral discretizations on the ball, with potential for efficient implementation using fast transforms.

Abstract

Our main objective in this work is to show how Sobolev orthogonal polynomials emerge as a useful tool within the framework of spectral methods for boundary-value problems. The solution of a boundary-value problem for a stationary Schrödinger equation on the unit ball can be studied from a variational perspective. In this variational formulation, a Sobolev inner product naturally arises. As test functions, we consider the linear space of the polynomials satisfying the boundary conditions on the sphere, and a basis of mutually orthogonal polynomials with respect to the Sobolev inner product is provided. The basis of the proposed method is given in terms of spherical harmonics and univariate Sobolev orthogonal polynomials. The connection formula between these Sobolev orthogonal polynomials and the classical orthogonal polynomials on the ball is established. Consequently, the Sobolev Fourier coefficients of a function satisfying the boundary value problem are recursively derived. Finally, one numerical experiment is presented.

A fully diagonalized spectral method on the unit ball

TL;DR

This work develops a fully diagonalized spectral method for Dirichlet boundary-value problems on the unit ball by leveraging Sobolev orthogonal polynomials. It builds a basis from spherical harmonics and univariate radial Sobolev polynomials, enabling direct computation of Sobolev-Fourier coefficients via a connection to classical ball polynomials and avoiding large linear systems. The approach introduces the univariate Sobolev polynomials and their connection to Jacobi polynomials, provides recursive schemes to obtain the needed coefficients, and demonstrates exponential convergence in a 2D numerical experiment. The method offers a scalable, high-accuracy alternative for spectral discretizations on the ball, with potential for efficient implementation using fast transforms.

Abstract

Our main objective in this work is to show how Sobolev orthogonal polynomials emerge as a useful tool within the framework of spectral methods for boundary-value problems. The solution of a boundary-value problem for a stationary Schrödinger equation on the unit ball can be studied from a variational perspective. In this variational formulation, a Sobolev inner product naturally arises. As test functions, we consider the linear space of the polynomials satisfying the boundary conditions on the sphere, and a basis of mutually orthogonal polynomials with respect to the Sobolev inner product is provided. The basis of the proposed method is given in terms of spherical harmonics and univariate Sobolev orthogonal polynomials. The connection formula between these Sobolev orthogonal polynomials and the classical orthogonal polynomials on the ball is established. Consequently, the Sobolev Fourier coefficients of a function satisfying the boundary value problem are recursively derived. Finally, one numerical experiment is presented.
Paper Structure (10 sections, 8 theorems, 102 equations, 3 figures, 1 table)

This paper contains 10 sections, 8 theorems, 102 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Spherical harmonics and orthogonal polynomials on the unit ball
  3. Classical Jacobi polynomials
  4. Orthogonal polynomials on the unit ball
  5. Sobolev orthogonal polynomials on the unit ball
  6. The radial parts
  7. An alternative algorithm
  8. The case $\kappa = 0$
  9. The approximate solution of the BVP
  10. A numerical experiment

Key Result

Lemma 2.1

For $\alpha > -1$ and $\beta > 0$,

Figures (3)

  • Figure 1: The solution $u(x_1,x_2)$ of the BVP.
  • Figure 2: A contour plot for the difference $u(x_1,x_2)-u_N(x_1,x_2)$ for $N=3$.
  • Figure 3: A logarithmic plot of the errors for $N = 0, 1, \ldots, 7$.

Theorems & Definitions (14)

  • Lemma 2.1
  • proof
  • Theorem 2.2
  • Proposition 3.1
  • proof
  • Proposition 4.1
  • proof
  • Proposition 4.2
  • proof
  • Proposition 4.3
  • ...and 4 more