Quantitative towers in finite difference calculus approximating the continuum

TL;DR

This paper develops finite-dimensional models that approximate multivector fields and differential forms, providing a combinatorial analysis of their differences from continuum structures, with explicit solutions in Euclidean space.

Contribution

It introduces consistent finite difference approximations for continuum differential geometric structures and analyzes their deviations combinatorially, offering explicit solutions in Euclidean settings.

Findings

Finite models approximate continuum structures with quantifiable differences.

Explicit combinatorial formulas are derived for Euclidean spaces.

The approach bridges discrete and continuum differential geometry.

Abstract

Multivector fields and differential forms at the continuum level have respectively two commutative associative products, a third composition product between them and various operators like $\partial$, $d$ and $*$ which are used to describe many nonlinear problems. The point of this paper is to construct consistent direct and inverse systems of finite dimensional approximations to these structures and to calculate combinatorially how these finite dimensional models differ from their continuum idealizations. In a Euclidean background there is an explicit answer which is natural statistically.