  The purpose of this paper is to analyze an efficient method for the solution
of the nonlinear system resulting from the discretization of the elliptic
Monge-Amp\`ere equation by a $C^0$ interior penalty method with Lagrange finite
elements. We consider the two-grid method for nonlinear equations which
consists in solving the discrete nonlinear system on a coarse mesh and using
that solution as initial guess for one iteration of Newton's method on a finer
mesh. Thus both steps are inexpensive. We give quasi-optimal $W^{1,\infty}$
error estimates for the discretization and estimate the difference between the
interior penalty solution and the two-grid numerical solution. Numerical
experiments confirm the computational efficiency of the approach compared to
Newton's method on the fine mesh.
