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In this study, we consider the nonlinear Sch\"odinger equation (NLS) with thezero-boundary condition on a two- or three-dimensional large finite cubiclattice. We prove that its solution converges to that of the NLS on the entireEuclidean space with simultaneous reduction in the lattice distance andexpansion of the domain. Moreover, we obtain a precise global-in-time bound forthe rate of convergence. Our proof heavily relies on Strichartz estimates on afinite lattice. A key observation is that, compared to the case of a latticewith a fixed size [Y. Hong, C. Kwak, S. Nakamura, and C. Yang, \emph{Finitedifference scheme for two-dimensional periodic nonlinear {S}chr\"{o}dingerequations}, Journal of Evolution Equations \textbf{21} (2021), no.~1,391--418.], the loss of regularity in Strichartz estimates can be reduced asthe domain expands, depending on the speed of expansion. This allows us toaddress the physically important three-dimensional case.

On the continuum limit for the discrete Nonlinear Schr\"odinger equation on a large finite cubic lattice