A radial basis function method for computing Helmholtz–Hodge decompositions. (27th June 2016)

• Record Type: Journal Article
• Authors: Fuselier, Edward J.; Wright, Grady B.
• Is Part Of: IMA journal of numerical analysis. Volume 37:Number 2(2017)
• Page Start: 774
• Page End: 797
• Publication Date: 2016-06-27
• Subjects: radial basis functions -- kernel methods -- vector decomposition -- divergence-free approximation -- curl-free approximation; Numerical analysis -- Periodicals; 519.405
• DOI: 10.1093/imanum/drw027
• Languages: English
• ISSNs: 0272-4979

Abstract

Abstract : A radial basis function method based on matrix-valued kernels is presented and analysed 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 the other where the tangential component of the curl-free part of the field is 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.