A rapid and well-conditioned algorithm for the Helmholtz--Hodge decomposition of vector fields on the sphere
Provides an efficient and numerically stable method for a fundamental operation in spherical vector field analysis, benefiting geophysics, computer graphics, and other domains.
The paper presents a fast, well-conditioned algorithm for Helmholtz-Hodge decomposition of vector fields on the sphere, achieving optimal complexity and low error growth with truncation degree.
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 $QR$ decompositions that factor and execute in optimal complexity. Rigorous upper bounds on the $2$-norm relative condition number of the banded linear systems support the observable low error growth with respect to truncation degree.