A Radial Basis Function Method for Computing Helmholtz-Hodge Decompositions
This work provides a novel meshless approach for vector field decomposition, addressing boundary condition handling for practitioners in computational physics and engineering.
The paper presents an RBF method using matrix-valued kernels for computing Helmholtz-Hodge decompositions on bounded domains, with boundary conditions imposed directly on vector fields. Error estimates and numerical examples demonstrate the method's effectiveness.
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.