This paper deals with the derivation and analysis of reduced order elliptic PDE models on fractured domains. We use a Fourier analysis to obtain coupling conditions between subdomains, when the fracture is represented as a hypersurface embedded in the surrounded rock matrix. We compare our results to prominent examples from the literature, for diffusive models. In a second step, we present error estimates for the reduced order models in terms of the fracture width. For the proofs, we rely on a combination of Fourier analysis, asymptotic expansions and functional analysis. Finally, we study the behaviour of the error of the reduced order solutions on numerical test cases, when the fracture width tends to zero.