Tensor calculus in polar coordinates using Jacobi polynomials

TL;DR

This paper introduces a new spectral basis built from Jacobi polynomials for efficiently solving tensor PDEs in polar coordinates, ensuring regularity at the origin and extending to other geometries.

Contribution

It develops a comprehensive set of bases and operators for tensor PDEs in polar coordinates, maximizing linear operation bandedness and extending spectral methods to more general geometries.

Findings

Bases satisfy regularity at r=0 for all tensor fields

Operators form a Heisenberg algebra from polynomial relations

Method demonstrates high accuracy and ease of use in applications

Abstract

Spectral methods are an efficient way to solve partial differential equations on domains possessing certain symmetries. The utility of a method depends strongly on the choice of spectral basis. In this paper we describe a set of bases built out of Jacobi polynomials, and associated operators for solving scalar, vector, and tensor partial differential equations in polar coordinates on a unit disk. By construction, the bases satisfy regularity conditions at r=0 for any tensorial field. The coordinate singularity in a disk is a prototypical case for many coordinate singularities. The work presented here extends to other geometries. The operators represent covariant derivatives, multiplication by azimuthally symmetric functions, and the tensorial relationship between fields. These arise naturally from relations between classical orthogonal polynomials, and form a Heisenberg algebra. Other past work uses more specific polynomial bases for solving equations in polar coordinates. The main innovation in this paper is to use a larger set of possible bases to achieve maximum bandedness of linear operations. We provide a series of applications of the methods, illustrating their ease-of-use and accuracy.