A rapid and well-conditioned algorithm for the Helmholtz--Hodge decomposition of vector fields on the sphere

TL;DR

This paper introduces a fast, well-conditioned algorithm for Helmholtz--Hodge decomposition on the sphere, leveraging spherical harmonics and banded QR decompositions for optimal efficiency and stability.

Contribution

The paper presents a novel algorithm that uncouples spherical harmonic modes and solves resulting banded linear systems efficiently, improving stability and speed for Helmholtz--Hodge decomposition on the sphere.

Findings

Low error growth with increasing truncation degree

Optimal complexity in solving banded linear systems

Rigorous bounds on condition number ensure stability

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 $QR$ 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.