Generators of $H^1(\Gamma ,\partial {\Gamma }^c)$ with $\partial {\Gamma }^c\subset \partial \Gamma$ for Triangulated Surfaces $\Gamma$: Construction and Classification of Global Loops

Abstract

Given a compact surface $\Gamma$ embedded in ${\mathbb{R}}^3$ with boundary $\partial \Gamma$, our goal is to construct a set of representatives for a basis of the relative cohomology group $H^1(\Gamma ,\partial {\Gamma }^c)$, where ${\Gamma }^c$ is a specified subset of $\partial \Gamma$. To achieve this, we propose a novel graph-based algorithm with two key features: it is applicable to non-orientable surfaces, thereby generalizing the construction of Hiptmair and Ostrowski [SIAM J. Comput., 31 (2002)], and it has a worst-case time complexity that is linear in the number of edges of the mesh $\mathcal{K}$ triangulating $\Gamma$. Importantly, this algorithm serves as a critical pre-processing step to address the low-frequency breakdown encountered in boundary element discretizations of integral equation formulations.