The geometry of convergence in numerical analysis

TL;DR

This paper explores how topological approaches to convergence in numerical analysis generalize traditional metric-based concepts, focusing on the geometric structure of mesh functions and partial maps.

Contribution

It introduces a topological framework for understanding convergence in numerical analysis, extending beyond metric spaces to more general topological spaces.

Findings

Topologies on partial maps generalize classical convergence notions.

Mesh function domains are strict subsets of the space of variables.

Geometric interpretation of convergence in topological terms.

Abstract

The domains of mesh functions are strict subsets of the underlying space of continuous independent variables. Spaces of partial maps between topological spaces admit topologies which do not depend on any metric. Such topologies geometrically generalize the usual numerical analysis definitions of convergence.