  We study the asymptotical compatibility of the Fourier spectral method in
multidimensional space for the Nonlocal Ohta-Kawasaka (NOK) model, which is
proposed in our previous work. By introducing the Fourier collocation
discretization for the spatial variable, we show that the asymptotical
compatibility holds in 2D and 3D over a periodic domain. For the temporal
discretization, we adopt the second-order backward differentiation formula
(BDF) method. We prove that for certain nonlocal kernels, the proposed time
discretization schemes inherit the energy dissipation law. In the numerical
experiments, we verify the asymptotical compatibility, the second-order
temporal convergence rate, and the energy stability of the proposed schemes.
More importantly, we discover a novel square lattice pattern when certain
nonlocal kernel are applied in the model. In addition, our numerical
experiments confirm the existence of an upper bound for the optimal number of
bubbles in 2D for some specific nonlocal kernels. Finally, we numerically
explore the promotion/demotion effect induced by the nonlocal horizon, which is
consistent with the theoretical studies presented in our earlier work.
