Computing with functions in the ball. (English) Zbl 1452.65001
Summary: 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 decompositions of a vector field defined on the ball.
MSC:
| 65-04 | Software, source code, etc. for problems pertaining to numerical analysis |
| 65Dxx | Numerical approximation and computational geometry (primarily algorithms) |
Keywords:
double Fourier sphere method; poloidal-toroidal decomposition; Helmholtz-Hodge decompositionReferences:
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