Computing with functions in the ball

TL;DR

This paper introduces algorithms in MATLAB for high-precision numerical computation with smooth functions on the unit ball, enabling efficient operations like evaluation, differentiation, and solving PDEs using a 3D Fourier sphere method.

Contribution

It presents a novel object-oriented MATLAB framework for adaptive, high-precision function computations on the ball, including new algorithms for vector calculus operations.

Findings

Functions are resolved to machine precision using a 3D Fourier sphere method.

Efficient algorithms for vector calculus operations on the ball.

Implementation of Helmholtz and Hodge decompositions for vector fields.

Abstract

A collection of algorithms in object-oriented MATLAB is described for numerically computing with smooth functions defined on the unit ball in the Chebfun software. Functions are numerically and adaptively resolved to essentially machine precision by using a three-dimensional analogue of the double Fourier sphere method to form "ballfun" objects. Operations such as function evaluation, differentiation, integration, fast rotation by an Euler angle, and a Helmholtz solver are designed. Our algorithms are particularly efficient for vector calculus operations, and we describe how to compute the poloidal-toroidal and Helmholtz--Hodge decomposition of a vector field defined on the ball.