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