A Reconstructed Discontinuous Approximation to Monge-Ampère Equation in Least Square Formulation (Advances in Applied Mathematics and Mechanics, 2023)
We propose a numerical method to solve the Monge-Ampère 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 thefirst step, we solve a nonlinear system to obtain the approximation to the gradient.A Newton iteration is adopted to handle the nonlinearity of the system. The approximation to the primitive variable is obtained from the approximate gradient bya trivial least squares finite element method in the second step. Numerical examplesin both two and three dimensions are presented to show an optimal convergencerate in accuracy. It is interesting to observe that the approximation solution is piecewise convex. Particularly, with the reconstructed approximation space, the proposedmethod numerically demonstrates a remarkable robustness. The convergence of theNewton iteration does not rely on the initial values. The dependence of the convergence on the penalty parameter in the discretization is also negligible, in comparisonto the classical discontinuous approximation space.