On the differential geometry of numerical schemes and weak solutions of functional equations

TL;DR

This paper explores the differential geometric structures underlying numerical schemes for functional equations, providing a unified framework that clarifies the smooth dependence of solutions on parameters and reinterprets classical proofs.

Contribution

It introduces a differential geometric framework for numerical methods involving Cauchy sequences, including a new tangent space concept and a Cauchy diffeology, unifying existing proofs.

Findings

Reinterpretation of the implicit functions theorem without norm estimates

Smooth dependence of solutions on triangulation parameters in finite element methods

Development of Cauchy diffeology and generalized tangent spaces for Cauchy sequence spaces

Abstract

We exhibit differential geometric structures that arise in numerical methods, based on the construction of Cauchy sequences, that are currently used to prove explicitly the existence of weak solutions to functional equations. We describe the geometric framework, highlight several examples and describe how two well-known proofs fit with our setting. The first one is a re-interpretation of the classical proof of an implicit functions theorem in an ILB setting, for which our setting enables us to state an implicit functions theorem without additional norm estimates, and the second one is the finite element method of the Dirichlet problem where the set of triangulations appear as a smooth set of parameters. In both case, smooth dependence on the set of parameters is established. Before that, we develop the necessary theoretical tools, namely the notion of Cauchy diffeology on spaces of Cauchy sequences and a new generalization of the notion of tangent space to a diffeological space.