  In this article, we present and analyze a finite element numerical scheme for
a three-component macromolecular microsphere composite (MMC) hydrogel model,
which takes the form of a ternary Cahn-Hilliard-type equation with
Flory-Huggins-deGennes energy potential. The numerical approach is based on a
convex-concave decomposition of the energy functional in multi-phase space, in
which the logarithmic and the nonlinear surface diffusion terms are treated
implicitly, while the concave expansive linear terms are explicitly updated. A
mass lumped finite element spatial approximation is applied, to ensure the
positivity of the phase variables. In turn, a positivity-preserving property
can be theoretically justified for the proposed fully discrete numerical
scheme. In addition, unconditional energy stability is established as well,
which comes from the convexity analysis. Several numerical simulations are
carried out to verify the accuracy and positivity-preserving property of the
proposed scheme.
