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

Abstract

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 inverse limit of branch 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 a Dirichlet problem where the set of triangulations appears as a smooth set of parameters. In both cases, 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.