A spectral method based on $(0,2)$ Jacobi polynomials. Application to Poisson problems in a sphere

TL;DR

This paper introduces a spectral method utilizing $(0,2)$ Jacobi polynomials tailored for spherical Poisson problems, emphasizing their orthogonality and quadrature properties, and demonstrates its effectiveness through numerical tests.

Contribution

The paper develops a novel spectral method based on $(0,2)$ Jacobi polynomials for spherical Poisson problems, including new quadrature results and a discrete transform.

Findings

Effective spectral method for Poisson problems in a sphere

New quadrature and discrete transform results for Jacobi polynomials

Numerical validation using C++ library Lorene

Abstract

A new spectral method is built resorting to $(0,2)$ Jacobi polynomials. We describe the origin and the properties of these polynomials. This choice of polynomials is motivated by their orthogonality properties with the respect to the weight $r^2$ used in spherical geometry. New results about Jacobi-Gauss-Lobatto quadratures are proven, leading to a discrete Jacobi transform. Numerical tests for Poisson problems in a sphere are presented using the C++ library \textsc{lorene}.