  We consider discrete nonlinear Schr\"odinger equations (DNLS) on the lattice
$h\mathbb{Z}^d$ whose linear part is determined by the discrete Laplacian which
accounts only for nearest neighbor interactions, or by its fractional power. We
show that in the continuum limit $h\to 0$, solutions to DNLS converge strongly
in $L^2$ to those to the corresponding continuum equations, but a precise rate
of convergence is also calculated. In particular cases, this result improves
weak convergence in Kirkpatrick, Lenzmann and Staffilani \cite{KLS}. Our proof
is based on a suitable adjustment of dispersive PDE techniques to a discrete
setting. Notably, we employ uniform-in-$h$ Strichartz estimates for discrete
linear Schr\"odinger equations in \cite{HY}, which quantitatively measure
dispersive phenomena on the lattice. Our approach could be adapted to a more
general setting like \cite{KLS} as long as the desired Strichartz estimates are
obtained.

Analysis of PDEs