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In this paper, we develop a new FEM-based method to solve the problem of scattering from an infinite periodic array of identical cavities engraved in an infinite perfect electric conductor screen. The FEM formulation is applied inside only one cavity to derive a linear system of equations associated with the nodal field values within the cavity. The surface integral equation using the free-space Green's function is applied at the openings of all cavities as a global boundary condition. Taking advantage of field's periodicity at the apertures of the cavities, the free space Green's function is replaced by the quasi-periodic Green's function, thus limiting the surface integral equation to the aperture of only one cavity. The advantage of this method is that no periodic boundary condition which would require a constrained mesh scheme is used in this formulation. Also we emphasize that in the method presented here, the surface integral equation using the quasi-periodic Green's function is used to derive a linear system of equation as a constraint which connects the field values on the boundary to the field values on the apertures of the single cavity in the array by considering the coupling between all cavities.