Manifold Harmonics and Their Application to Computational Electromagnetics

Abstract

The eigenfunctions of the Laplace-Beltrami operator (LBO), or manifold harmonic basis (MHB), have many applications in mathematical physics, differential geometry, machine learning, and topological data analysis. MHB allows us to associate a frequency spectrum to a function on a manifold, analogous to the Fourier decomposition. This insight can be used to build a framework for analysis. The purpose of this paper is to review and illustrate such possibilities for computational electromagnetics as well as chart a potential path forward. To this end, we introduce three features of MHB: (a) enrichment for analysis of multiply connected domains, (b) local enrichment (L-MHB) and (c) hierarchical MHB (H-MHB) for reuse of data from coarser to fine geometry discretizations. Several results highlighting the efficacy of these methods are presented.