  We present and analyze an unconditionally energy stable and convergent finite
difference scheme for the Functionalized Cahn-Hilliard equation. One key
difficulty associated with the energy stability is based on the fact that one
nonlinear energy functional term in the expansion appears as non-convex,
non-concave. To overcome this subtle difficulty, we add two auxiliary terms to
make the combined term convex, which in turns yields a convex-concave
decomposition of the physical energy. As a result, an application of the convex
splitting methodology assures both the unique solvability and the unconditional
energy stability of the proposed numerical scheme. To deal with a 4-Laplacian
solver in an $H^{-1}$ gradient flow at each time step, we apply an efficient
preconditioned steepest descent algorithm to solve the corresponding nonlinear
systems. In addition, a global in time $H_{\rm per}^2$ stability of the
numerical scheme is established at a theoretical level, which in turn ensures
the full order convergence analysis of the scheme. A few numerical results are
presented, which confirm the stability and accuracy of the proposed numerical
scheme.
