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.

中文翻译：

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数值格式的微分几何与泛函方程的弱解

我们展示了数值方法中出现的微分几何结构，基于柯西序列的构造，目前用于明确证明函数方程弱解的存在。我们描述了几何框架，突出显示了几个例子，并描述了两个众所周知的证明如何与我们的设置相适应。第一个是对 ILB 设置中隐函数定理的经典证明的重新解释，我们的设置使我们能够在没有额外范数估计的情况下陈述隐函数定理，第二个是有限元方法Dirichlet 问题，其中三角剖分集显示为一组平滑的参数。在这两种情况下，都建立了对参数集的平滑依赖。在此之前，我们开发了必要的理论工具，