  Integrable and nonintegrable discrete nonlinear Schr\"odinger equations (NLS)
are significant models to describe many phenomena in physics. Recently,
Ablowitz and Musslimani introduced a class of reverse space, reverse time and
reverse space-time nonlocal integrable equations, including nonlocal NLS,
nonlocal sine-Gordon equation and nonlocal Davey-Stewartson equation etc. And,
the integrable nonlocal discrete NLS has been exactly solved by inverse
scattering transform. In this paper, we study a nonintegrable discrete nonlocal
NLS which is direct discretization version of the reverse space nonlocal NLS.
By applying discrete Fourier transform and modified Neumann iteration, we
present its stationary solutions numerically. The linear stability of the
stationary solutions is examined. Finally, we study the Cauchy problem for
nonlocal NLS equation numerically and find some different and new properties on
the numerical solutions comparing with the numerical solutions of the Cauchy
problem for NLS equation.
