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.

翻译：在本研究中,我们考虑的是非线性Sch\'odinger方程式(NLS),该方程式的零边界条件为两维或三维的大型立方体。我们证明,该方程式的解决方案与整个欧几里德空间的NLS的解决方案一致,同时缩短了拉特斯距离和扩展域。此外,我们获得了精确的全时趋同速度。我们的证据严重依赖Strichartz对有限拉蒂的估算。一个关键的观察是,与固定大小的拉特斯相比[Y. Hong, C. Kwak, S. Nakamura和C. Yang,\emph{Finite differite progy progations of 双维的周期非线性非线性定期非线性平衡 {S} ({chr\}}}}{odirdinger complas},《进化方程学杂志》\ textbf} (2021, no. b. 391--418.], Strichartz估计的规律性损失可以随着地域扩展范围的扩展范围扩展速度扩大而减少。