  Solitons of the purely cubic nonlinear Schr\"odinger equation in a space
dimension of $n \geq 2$ suffer critical and supercritical collapses. These
solitons can be stabilized in a cubic-quintic nonlinear medium. In this paper,
we analyze the Crank-Nicolson finite difference scheme for the (2+1)D
cubic-quintic nonlinear Schr\"odinger equation with cubic damping. We show that
both the discrete solution, in the discrete $L^2$-norm, and discrete energy are
bounded. By using appropriate settings and estimations, the existence and the
uniqueness of the numerical solution are proved. In addition, the error
estimations are established in terms of second order for both space and time in
discrete $L^2$-norm and $H^1$-norm. Numerical simulations for the (2+1)D
cubic-quintic nonlinear Schr\"odinger equation with cubic damping are conducted
to validate the convergence.
