A Meshfree Method for Solving the Monge-Ampère Equation

Abstract

This paper solves the two-dimensional Dirichlet problem for the Monge-Ampère equation by a strong meshless collocation technique that uses a polynomial trial space and collocation in the domain and on the boundary. Convergence rates may be up to exponential, depending on the smoothness of the true solution, and this is demonstrated numerically and proven theoretically, applying a sufficiently fine collocation discretization. A much more thorough investigation of meshless methods for fully nonlinear problems is in preparation.