A fully diagonalized spectral method on the unit ball
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.
Cite
@article{arxiv.2601.15911,
title = {A fully diagonalized spectral method on the unit ball},
author = {Miguel A. Piñar},
journal= {arXiv preprint arXiv:2601.15911},
year = {2026}
}
Comments
This preprint has not undergone peer review or any post-submission improvements or corrections. The Version of Record of this article is published in Numerical Algorithms, and is available online at https://doi.org/10.1007/s11075-026-02315-w