A Cubic Whitney and Further Developments in Geometric Discretisation

TL;DR

This paper extends geometric discretisation to cubic complexes, incorporating metric considerations through volume factors and modified inner products, enhancing the theoretical framework for discrete geometric analysis.

Contribution

It generalizes geometric discretisation to cubic complexes and introduces metric integration via volume factors and inner product modifications.

Findings

Extended the theory to cubic complexes.

Incorporated metric via volume factors.

Modified inner product for better discretisation.

Abstract

Geometric discretisation draws analogies between discrete objects and operations on a complex with continuum ones on a manifold. We generalise the theory to the cubic case and incorporate metric, by adding volume factors to our discrete Hodge star and then by modifying our inner product which leads to the same result.