A new finite element technique for calculating the equilibrium configuration of atomic structures called the Consistent Atomic-scale Finite Element (CAFÉ) is introduced. Unlike traditional approaches for linking the atomic structure to its equivalent continuum, this method directly connects the atomic degrees of freedom to a reduced set of finite element degrees of freedom without passing through an intermediate homogenized continuum. As a result, there is no need to introduce stress and strain measures at the atomic level. The Tersoff-Brenner interatomic potential is used to calculate the consistent tangent stiffness matrix of the structure. In this finite element formulation, all local and non-local interactions between carbon atoms are taken into account using overlapping finite elements. In addition, a consistent hierarchical finite element modeling technique is developed for adaptively coarsening and refining the mesh over different parts of the model. This process is consistent with the underlying atomic structure and, by refining the mesh, molecular mechanic results will be recovered. This method is valid across the scales and can be used to concurrently model atomistic and continuum phenomena. Using this technique, vibration frequencies of carbon nanostructures such as graphene sheet and carbon nanotube are studied and results are compared with those using equivalent homogenized continuum. Range of the applicability of the continuum approach to the vibration of these nanostructures is discussed. Furthermore, nonlinear mechanics of radial deformation of a single-walled carbon nanotube and inversion of a carbon nanocone are studied.