  Computing stationary states is an important topic for phase field crystal
(PFC) models. Great efforts have been made for energy dissipation of the
numerical schemes when using gradient flows. However, it is always
time-consuming due to the requirement of small effective time steps. In this
paper, we propose an adaptive accelerated proximal gradient method for finding
the stationary states of PFC models. The energy dissipation is guaranteed and
the convergence property is established for the discretized energy functional.
Moreover, the connections between generalized proximal operator with classical
(semi-)implicit and explicit schemes for gradient flow are given. Extensive
numerical experiments, including two three dimensional periodic crystals in
Landau-Brazovskii (LB) model and a two dimensional quasicrystal in
Lifshitz-Petrich (LP) model, demonstrate that our approach has adaptive time
steps which lead to significant acceleration over semi-implicit methods for
computing complex structures. Furthermore, our result reveals a deep physical
mechanism of the simple LB model via which the sigma phase is first discovered.
