A fully diagonalized spectral method on the unit ball

TL;DR

This paper introduces a spectral method on the unit ball using Sobolev orthogonal polynomials, enabling efficient boundary-value problem solutions for the Schrödinger equation with a focus on basis construction and recursive coefficient computation.

Contribution

It develops a fully diagonalized spectral method based on Sobolev orthogonal polynomials, connecting them with classical polynomials and providing recursive formulas for Fourier coefficients.

Findings

Successful formulation of a spectral method on the unit ball

Derivation of recursive formulas for Sobolev Fourier coefficients

Implementation demonstrated through a numerical experiment

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 Schr\"odinger 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.