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We propose a numerical method to solve the Monge-Ampere equation which admitsa classical convex solution. The Monge-Ampere equation is reformulated into anequivalent first-order system. We adopt a novel reconstructed discontinuousapproximation space which consists of piecewise irrotational polynomials. Thisspace allows us to solve the first order system in two sequential steps. In thefirst step, we solve a nonlinear system to obtain the approximation to thegradient. A Newton iteration is adopted to handle the nonlinearity in thesystem. The approximation to the primitive variable is obtained from theapproximate gradient by a trivial least squares finite element method in thesecond step. Numerical examples in both two dimensions and three dimensions arepresented to show an optimal convergence rate in accuracy. It is interesting toobserve that the approximate solution is piecewise convex in each element.Particularly, with the reconstructed approximation space, the proposed methoddemonstrates a remarkable robustness. The convergence of the Newton iterationdoes not rely on the initial values, which shows a very different behaviourfrom references. The dependence of the convergence on the penalty parameter inthe discretization is also negligible, in comparison to the classicaldiscontinuous approximation space.

A Reconstructed Discontinuous Approximation to Monge-Ampere Equation in Least Square Formation