A rapid and well-conditioned algorithm for the Helmholtz--Hodge decomposition of vector fields on the sphere
arXiv:1809.04555
Abstract
A rapid algorithm is derived for the Helmholtz--Hodge decomposition on the surface of the sphere in spherical coordinates. The algorithm uncouples modes of spherical harmonics with different absolute order, writes the conversion as barely-overdetermined banded linear systems, and solves them with banded decompositions that factor and execute in optimal complexity. Rigorous upper bounds on the -norm relative condition number of the banded linear systems support the observable low error growth with respect to truncation degree.