Numerous models encountered in science and engineering remain nowadays , despite the impressive recent progresses attained in computational simulation techniques, intractable when the usual and well experienced discretization techniques are applied for their numerical simulation. Thus, different challenging issues are waiting for the proposal of new alternative advanced simulation techniques. Separated representations offer the possibility to address some challenging models with CPU time savings of some orders of magnitude. In other cases, they allowed to address models never until now solved. The number of published works concerning this kind of approximation remains quite reduced, and then, numerous difficulties that were successfully circumvented in the context of more experienced discretization techniques, as is the case of the finite element method, must be considered again within the separated representation framework. One of these issues in the one that concerns the treatment of localized behavior of model solutions. This work focuses on this topic and propose an efficient finite element (or extended finite element) enrichment of usual separated representation.