A Radial Basis Function Method for Computing Helmholtz-Hodge Decompositions

TL;DR

This paper introduces a matrix-valued radial basis function method for computing Helmholtz-Hodge decompositions on bounded domains, allowing boundary conditions to be imposed directly on vector fields and providing error estimates with numerical validation.

Contribution

It presents a novel RBF-based approach that directly imposes boundary conditions on vector fields for Helmholtz-Hodge decompositions, differing from existing methods that use potentials.

Findings

Method achieves accurate decompositions from scattered data.

Provides Sobolev-type error estimates for the decompositions.

Numerical examples validate the theoretical results.

Abstract

A radial basis function (RBF) method based on matrix-valued kernels is presented and analyzed for computing two types of vector decompositions on bounded domains: one where the normal component of the divergence-free part of the field is specified on the boundary, and one where the tangential component of the curl-free part of the field specified. These two decompositions can then be combined to obtain a full Helmholtz-Hodge decomposition of the field, i.e. the sum of divergence-free, curl-free, and harmonic fields. All decompositions are computed from samples of the field at (possibly scattered) nodes over the domain, and all boundary conditions are imposed on the vector fields, not their potentials, distinguishing this technique from many current methods. Sobolev-type error estimates for the various decompositions are provided and demonstrated with numerical examples.