  We propose a numerical method to solve the Monge-Ampere equation which admits
a classical convex solution. The Monge-Ampere equation is reformulated into an
equivalent first-order system. We adopt a novel reconstructed discontinuous
approximation space which consists of piecewise irrotational polynomials. This
space allows us to solve the first order system in two sequential steps. In the
first step, we solve a nonlinear system to obtain the approximation to the
gradient. A Newton iteration is adopted to handle the nonlinearity in the
system. The approximation to the primitive variable is obtained from the
approximate gradient by a trivial least squares finite element method in the
second step. Numerical examples in both two dimensions and three dimensions are
presented to show an optimal convergence rate in accuracy. It is interesting to
observe that the approximate solution is piecewise convex in each element.
Particularly, with the reconstructed approximation space, the proposed method
demonstrates a remarkable robustness. The convergence of the Newton iteration
does not rely on the initial values, which shows a very different behaviour
from references. The dependence of the convergence on the penalty parameter in
the discretization is also negligible, in comparison to the classical
discontinuous approximation space.
