A Meshfree Method for Solving the Monge-Ampère Equation
It provides a new numerical approach for fully nonlinear PDEs, but the work is preliminary and incremental.
The paper presents a meshless collocation method for solving the 2D Monge-Ampère equation, achieving up to exponential convergence rates both numerically and theoretically.
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.