In the literature, FEM has sometimes been characterized as a local approach, but IMO this needs to be corrected.
The piecewise continuous trial-functions of FEM can be looked at from two different viewpoints:
(i) If FEM is seen as an expansion method making use of basis functions, then naturally the comparison is with the Fourier-theoretic approaches (and all the derived, consequent or similar ones). The basis functions for the latter are global in the sense they have supports all over the domain. This, indeed, is unlike the limited (piecewise) support of the FEM trial-functions.