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